MATHEMATICAL    MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN   and    ROBERT   S.   WOODWARD. 


No.  13. 


THE    THEORY 


OF 


NUMBERS 


BY 

ROBERT    D.   CARMICHAEL, 

Associate  Professor  of  Mathematics  in  Indiana  University 


FIRST    EDITION 
FIRST    THOUSAND 


NEW  YORK 

JOHN   WILEY  &  SONS,  Inc. 

London:   CHAPMAN  &  HALL:   Limited 

1914 


Copyright,  19 14. 

BY 

ROBERT  D.  CARMICHAEL. 

$0*0*13 


THE    SCIENTIFIC    PRESS 

ROBERT    DRUMMOND    AND   COMPANY 

BROOKLYN,   N.    Y. 


EDITORS'  PREFACE. 


The  volume  called  Higher  Mathematics,  the  third  edition 
of  which  was  published  in  1900,  contained  eleven  chapters  by- 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  was  discontinued  in 
1906,  and  the  chapters  have  since  been  issued  in  separate 
Monographs,  they  being  generally  enlarged  by  additional 
articles  or  appendices  which  either  amplify  the  former  pres- 
entation or  record  recent  advances.  This  plan  of  publication 
was  arranged  in  order  to  meet  the  demand  of  teachers  and 
the  convenience  of  classes,  and  it  was  also  thought  that  it 
would  prove  advantageous  to  readers  in  special  lines  of  mathe- 
matical literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  demand 
seems  to  warrant  it.  Among  the  topics  which  are  under  con- 
sideration are  those  of  elliptic  functions,  the  theory  of  quantics, 
the  group  theory,  the  calculus  of  variations,  and  non-Euclidean 
geometry;  possibly  also  monographs  on  branches  of  astronomy, 
mechanics,  and  mathematical  physics  may  be  included.  It  is 
the  hope  of  the  editors  that  this  Series  of  Monographs  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 


302093 


PREFACE 


The  purpose  of  this  little  book  is  to  give  the  reader  a  con- 
venient introduction  to  the  theory  of  numbers,  one  of  the  most 
extensive  and  most  elegant  disciplines  in  the  whole  body  of 
mathematics.  The  arrangement  of  the  material  is  as  follows : 
The  first  five  chapters  are  devoted  to  the  development  of  those 
elements  which  are  essential  to  any  study  of  the  subject.  The 
sixth  and  last  chapter  is  intended  to  give  the  reader  some 
indication  of  the  direction  of  further  study  with  a  brief  account 
of  the  nature  of  the  material  in  each  of  the  topics  suggested. 
The  treatment  throughout  is  made  as  brief  as  is  possible  con- 
sistent with  clearness  and  is  confined  entirely  to  fundamental 
matters.  This  is  done  because  it  is  believed  that  in  this  way 
the  book  may  best  be  made  to  serve  its  purpose  as  an  intro- 
duction to  the  theory  of  numbers. 

Numerous  problems  are  supplied  throughout  the  text. 
These  have  been  selected  with  great  care  so  as  to  serve  as  excel- 
lent exercises  for  the  student's  introductory  training  in  the 
methods  of  number  theory  and  to  afford  at  the  same  time  a 
further  collection  of  useful  results.  The  exercises  marked  with 
a  star  are  more  difficult  than  the  others;  they  will  doubtless 
appeal  to  the  best  students. 

Finally,  I  should  add  that  this  book  is  made  up  from  the 
material  used  by  me  in  lectures  in  Indiana  University  during 
the  past  two  years;  and  the  selection  of  matter,  especially  of 
exercises,  has  been  based  on  the  experience  gained  in  this  way. 

R.  D.  Carmichael. 
4 


CONTENTS 


CHAPTER  I.     ELEMENTARY  PROPERTIES  OF  INTEGERS 

PAGE 

j  i.  Fundamental  Notions  and  Laws 7 

j  2.  Definition  of  Divisibility.     The  Unit 8 

\  3.  Prime  Numbers.    The  Sieve  of  Eratosthenes 10 

&  4.  The  Number  of  Primes  is  Infinite. 12 

&  5.  The  Fundamental  Theorem  of  Euclid 13 

I  6.  Divisibility  by  a  Prime  Number 13 

§  7.  The  Unique  Factorization  Theorem 14 

I  8.  The  Divisors  of  an  Integer 16 

§  9.  The  Greatest  Common  Factor  of  Two  or  More  Integers 18 

10.  The  Least  Common  Multiple  of  Two  or  More  Integers 20 

11.  Scales  of  Notation 22 

12.  Highest  Power  of  a  Prime  p  Contained  in  n\ 24 

13.  Remarks  Concerning  Prime  Numbers.  . '. 28 

CHAPTER  II.  ON  THE  INDICATOR  OF  AN  INTEGER 

14.  Definition.     Indicator  of  a  Prime  Power 30 

15.  The  Indicator  of  a  Product 30 

16.  The  Indicator  of  any  Positive  Integer 32 

17.  Sum  of  the  Indicators  of  the  Divisors  of  a  Number 35 

CHAPTER  III.     ELEMENTARY  PROPERTIES  OF  CONGRUENCES 

18.  Congruences  Modulo  m 37 

19.  Solutions  of  Congruences  by  Triai 39 

20.  Properties  of  Congruences  Relative  to  Division 40 

21.  Congruences  with  a  Prime  Modulus 41 

22.  Linear  Congruences 43 

CHAPTER  IV.     THE  THEOREMS  OF  FERMAT  AND  WILSON 

23.  Fermat's  General  Theorem 47 

24.  Euler's  Proof  of  the  Simple  Fermat  Theorem 48 

:  25.  Wilson's  Theorem 49 

5 


D  CONTENTS 

PAGE 

§  26.  The  Converse  of  Wilson's  Theorem 50 

§  27.  Impossibility  of  1  -2  -3  •  ...  •«  —  i-f  i=m*,  «>s 51 

§  28.  Extension  of  Fermat's  Theorem : 52 

§  29.  On  the  Converse  of  Fermat's  Simple  Theorem 55 

§  30.  Application  of  Previous  Results  to  Linear  Congruences 56 

§31.  Application  of  the  Preceding  Results  to  the  Theory  of  Quad- 
ratic Residues 57 

CHAPTER  V.    PRIMITIVE  ROOTS  MODULO  m 

§  32.  Exponent  of  an  Integer  Modulo  m 61 

§  i2>-  Another  Proof  of  Fermat's  General  Theorem 63 

§  34.  Definition  of  Primitive  Roots 65 

§  35.  Primitive  Roots  Modulo  p 66 

§  36.  Primitive  Roots  Modulo  pa,  p  an  Odd  Prime '. 68 

§  37.  Primitive  Roots  Modulo  2pa,  p  an  Odd  Prime 70 

§  38.  Recapitulation 71 

§  39.  Primitive  X-roots 71 

CHAPTER  VI.     OTHER  TOPICS 

§  40.  Introduction 76 

§  41.  Theory  of  Quadratic  Residues 77 

§  42.  Galois  Imaginaries 80 

§  43.  Arithmetic  Forms 81 

§  44.  Analytical  Theory  of  Numbers 83 

§  45.  Diophantine  Equations 84 

§  46.  Pythagorean  Triangles 85 

§  47.  The  Equation  xn+yn  =  in 9i 


THE  THEORY  OF  NUMBERS 


CHAPTER  I 
ELEMENTARY  PROPERTIES  OF  INTEGERS 

§  i.  Fundamental  Notions  and  Laws 

In  the  present  chapter  we  are  concerned  primarily  with 
certain  elementary  properties  of  the  positive  integers  i,  2,  3, 
4,  '...•.  .  It  will  sometimes  be  convenient,  when  no  confusion 
can  arise,  to  employ  the  word  integer  or  the  word  number  in 
the  sense  of  positive  integer. 

We  shall  suppose  that  the  integers  are  already  defined, 
either  by  the  process  of  counting  or  otherwise.  We  assume 
further  that  the  meaning  of  the  terms  greater,  less,  equal,  sum, 
difference,  product  is  known. 

From  the  ideas  and  definitions  thus  assumed  to  be  known 
follow  immediately  the  theorems : 

I.   The  sum  of  any  two  integers  is  an  integer. 
II.   The  difference  of  any  two  integers  is  an  integer. 
III.  The  product  of  any  two  integers  is  an  integer. 

Other  fundamental  theorems,  which  we  take  without  proof, 
are  embodied  in  the  following  formulas: 

IV.  a+b  =  b+a. 

V.  aXb  =  bXa. 

VI.  (a+b)+c  =  a+(b+c). 

VII.  (aXb)Xc  =  aX(bXc). 

VIII.  aX(b+c)  =aXb+aXc. 

Here  a,  b}  c  denote  any  positive  integers. 


8  ♦/•  *  2/2  •••{   ;  •:  XHIJO^y   Of   NUMBERS 

These  formulas  are  equivalent  in  order  to  the  following 
five  theorems:  addition  is  commutative;  multiplication  is 
commutative;  addition  is  associative;  multiplication  is  asso- 
ciative;  multiplication  is  distributive  with  respect  to  addition. 

EXERCISES 

Prove  the  following  relations: 


(1+2+  .  .  .  +n)K 


of  each  of  the  following  series:  \\ 

i3+33  +  53+  •  •  •  +(2w-i)3. 

5.  Discover  and  establish  the  law  suggested  by  the  equations  i2  =  0+1,  22  =  i-f-3, 
32  =  3+6,  42  =  6+io,  .  .  .;  by  the  equations  i  =  i3,  3+5  — 23,  7+9+11=3?, 
i3  +  i5  +  i74-i9  =  43, 

§  2.  Definition  of  Divisibility.     The  Unit 

Definitions.  An  integer  a  is  said  to  be  divisible  by  an 
integer  b  if  there  exists  an  integer  c  such  that  a  —  be.  It  is  clear 
from  this  definition  that  a  is  also  divisible  by  c.  The  integers 
b  and  c  are  said  to  be  divisors  or  factors  of  a;  and  a  is  said  to 
be  a  multiple  of  b  or  of  c.  The  process  of  finding  two  integers 
b  and  c  such  that  be  is  equal  to  a  given  integer  a  is  called  the 
process  of  resolving  a  into  factors  or  of  factoring  a;  and  a  is 
said  to  be  resolved  into  factors  or  to  be  factored. 

We  have  the  following  fundamental  theorems: 

I.  If  b  is  a  divisor  of  a  and  c  is  a  divisor  of  b,  then  c  is  a 
divisor  of  a. 

Since  &  is  a  divisor  of  a  there  exists  an  integer  /3  such  that 
a  =  &j8.  Since  c  is  a  divisor  of  b  there  exists  an  integer  7  such 
that  b  =  cy.  Substituting  this  value  of  b  in  the  equation  a  =  5/3 
we  have  a  =  cyfi.  But  from  theorem  III  of  §  i  it  follows  that 
7jS  is  an  integer;  hence,  c  is  a  divisor  of  a,  as  was  to  be  proved. 


ELEMENTARY   PROPERTIES   OF   INTEGERS  9 

II.  If  c  is  a  divisor  of  both  a  and  b,  then  c  is  a  divisor  of  the 
sum  of  a  and  b. 

From  the  hypothesis  of  the  theorem  it  follows  that  integers 
a  and  /3  exist  such  that 

a  =  ca,     b  =  c(3. 
Adding,  we  have 

a  +  b  =  ca+cP  =  c(a-\-p)=c8, 

where  5  is  an  integer.     Hence,  c  is  a  divisor  oi0j-b. 

III.  If  c  is  a  divisor  of  both  a  and  b,  then  c  is  a  divisor  of  the 
difference  of  a  and  b. 

The  proof  is  analogous  to  that  of  the  preceding  theorem. 

DEFINITIONS.  If  a  and  b  are  both  divisiblsJby  c,  then  c 
is  said  to  be  a  common  divisor  or  a  common  fa»r«:  a  and  b. 
Every  two  integers  have  the  common  factor  ifHrTne  greatest 
integer  which  divides  both  a  ariU  b  is  called  the  greatest  common 
divisor  of  a  and  b.  More  generally,  we  define  in  a  similar  way 
a  common  divisor  and  the  greatest  common  divisor  of  n  integers 
0i,  a2,   .  .  .  ,  On. 

Definitions.  If  an  integer  a  is  a  multiple -of  each  of  two 
or  more  integers  it  is  called  a  common  multiple  of  these  integers. 
The  product  of  any  set  of  integers  is  a  common  multiple  of  the 
set.  The  least  integer  which  is  a  multiple  of  each  of  two  or 
more  integers  is  called  their  least  common  multiple. 

It  is  evident  that  the  integer  i  is  a  divisor  of  every  integer 
and  that  it  is  the  only  integer  which  has  this  property.  It  is 
called  the  unit. 

Definition.  Two  or  more  integers  which  have  no  common 
factor  except  i  are  said  to  be  prime  to  each  other  or  to  be  rela- 
tively prime. 

Definition.  If  a  set  of  integers  is  such  that  no  two  of 
them  have  a  common  divisor  besides  i  they  are  said  to  be  prime 
each  to  each. 

EXERCISES 

i.  Prove  that  n3  —  n  is  divisible  by  6  for  every  positive  integer  n. 

2.  If  the  product  of  four  consecutive  integers  is  increased  by  i  the  result 
is  a  square  number. 

,S.  Show  that  24n+2+i  has  a  factor  different  from  itself  and  i  when  n  is  a 
positive  integer. 


10  THEORY   OF   NUMBERS 


§  3.  Prime  Numbers.    The  Sieve  of  Eratosthenes 

Definition.  If  an  integer  p  is  different  from  1  and  has 
no  divisor  except  itself  and  1  it  is  said  to  be  a  prime  number 
or  to  be  a  prime. 

Definition.  An  integer  which  has  at  least  one  divisor 
other  than  itself  and  1  is  said  to  be  a  composite  number  or  to 
be  composite. 

All  integers  are  thus  divided  into  three  classes: 

1)  The  unity 

2)  Prime  numbers;  ^j 

3)  Composite  numbers. 

We  have  seen  that  the  first  class  contains  only  a  single 
number.  The  third  class  evidently  contains  an  infinitude  of 
numbers;  for,  it  contains  all  the  numbers  2?,  23,  24,  .  .  .  . 
In  the  next  section  we  shall  show  that  the  second  class  also 
contains  an  infinitude  of  numbers.  We  shall  now  show  that 
every  number  of  the  third  class  contains  one  of  the  second 
class  as  a  factor,  by  proving  the  following  theorem: 

I.  Every  integer  greater  than  1  has  a  prime  factor. 

Let  m  be  any  integer  which  is  greater  than  1.  We  have 
to  show  that  it  has  a  prime  factor.  If  m  is  prime  there  is  the 
prime  factor  m  itself.     If  m  is  not  prime  we  have 

w  =  mim2, 

where  m\  and  ni2  are  positive  integers  both  of  which  are  less  than 
m.  If  either  ni\  or  ni2  is  prime  we  have  thus  obtained  a  prime 
factor  of  m.     If  neither  of  these  numbers  is  prime,  then  write 

rn\=mrimf2,     m\>i,     w/2>i. 

Both  ih\  and  mr2  are  factors  of  m  and  each  of  them  is  less  than 
mi.  Either  we  have  now  found  in  m\  or  m'2  a  prime  factor 
of  m  or  the  process  can  be  continued  by  separating  one  of  these 
numbers  into  factors.  Since  for  any  given  m  there  is  evideix.  I 
only  a  finite  number  of  such  steps  possible,  it  is  clear  that  w 


ELEMENTARY   PROPERTIES   OF  INTEGERS  11 

must  finally  arrive  at  a  prime  factor  of  m.  From  this  conclu- 
sion the  theorem  follows  immediately. 

Eratosthenes  has  given  a  useful  means  of  finding  the  prime 
numbers  which  are  less  than  any  given  integer  m.  It  may  be 
described  as  follows: 

Every  prime  except  2  is  odd.  Hence  if  we  write  down  every 
odd  number  from  3  up  to  m  we  shall  have  in  the  list  every  prime 
less  than  m  except  2.  Now  3^s_a_prime.  Leave  it  in  the  list; 
but  beginning  to  count  from  3  strike  out  every  third  number 
in  the  list.  Thus  every  number  divisible  by  3,  except  3  itself, 
is  cancelled.  Then  begin  from  5  and  cancel  every  fifth  num- 
ber. Then  begin  from  the  next  uncancelled  number,  namely 
7,  and  strike  out  every  seventh  number.  Then  begin  from 
the  next  uncancelled  number,  namely  11,  and  strike  out  every 
eleventh  number.  Proceed  in  this  way  up  to  m.  The  uncan- 
celled numbers  remaining  will  be  the  odd  primes  not  greater 
than  m. 

It  is  obvious  that  this  process  of  cancellation  need  not  be 
carried  altogether  so  far  as  indicated;  for  if  p  is  a  prime  greater 
than  Vw,  the  cancellation  of  any  pth  number  from  p  will  be 
merely  a  repetition  of  cancellations  effected  by  means  of  another 
factor  smaller  than  p,  as  one  may  see  by  use  of  the  following 
theorem. 

II.  An  integer  m  is  prime  if  it  has  no  prime  factor  equal  to 
or  less  than  I,  wJiere  I  is  the  greatest  integer  whose  square  is 
equal  to  or  less  than  m. 

Since  m  has  no  prime  factor  less  than  /,  it  follows  from 
theorem  I  that  it  has  no  factor  but  unity  less  than  /.  Hence, 
if  m  is  not  prime  it  must  be  the  product  of  two  numbers  each 
greater  than  /;  and  hence  it  must  be  equal  to  or  greater  than 
(7-f  i)2.  This  contradicts  the  hypothesis  on  /;  and  hence 
we  conclude  that  m  is  prime. 

EXERCISE 

By  means  of  the  method  of  Eratosthenes  determine  the  primes  less  than 
200. 


12  THEORY   OF   NUMBERS 


§4.  The  Number  of  Primes  is  Infinite 

I.  The  number  of  primes  is  infinite. 

We  shall  prove  this  theorem  by  supposing  that  the  number 
of  primes  is  not  infinite  and  showing  that  this  leads  to  a  con- 
tradiction. If  the  number  of  primes  is  not  infinite  there  is  a 
greatest  prime  number,  which  we  shall  denote  by  p.  Then 
form  the  number 

iV  =  i-2-3-.  .  .-/>+!. 


Now  by  theorem  I  of  §  3  N  has  a  prime  divisor  q.  But  every 
non-unit  divisor  of  iV,  is  obviously  greater  than  p.  Hence  q 
is  greater  than  />,  in  contradiction  to  the  conclusion  that  p  is 
the  greatest  prime.  Thus  the  proof  of  the  theorem  is  complete- 
In  a  similar  way  we  may  prove  the  following  theorem: 
II.  Among  the  integers  of  the  arithmetic  progression  5,  II, 
17,  23,  ...  ,  there  is  an  infinite  number  of  primes. 

If  the  number  of  primes  in  this  sequence  is  not  infinite 
there  is  a  greatest  prime  number  in  the  sequence;  supposing 
that  this  greatest  prime  number  exists  we' shall  denote  it  by  p. 
Then  the  number  N, 

iV^i-2-3-.  .    -#-i, 

is  not  divisible  by  any  number  less  than  or  equal  to  p.  This 
number  N,  which  is  of  the  form  6»  —  1,  has  a  prime  factor. 
If  this  factor  is  of  the  form  6&  —  1  we  have  already  reached  a 
contradiction,  and  our  theorem  is  proved.  If  the  prime  is  of 
the  form  6^i-+-i  the  complementary  factor  is  of  the  form  6^2  — 1» 
Every  prime  factor  of  6&2--1  is  greater  than  p.  Hence  we 
may  treat  6^2  —  1  as  we  did  6n  —  1 ,  and  with  a  like  result.  Hence 
we  must  ultimately  reach  a  prime  factor  of  the  form  6^3  —  1; 
for,  otherwise,  we  should  have  6w  — 1  expressed  as  a  product 
of  prime  factors  all  of  the  form  6/+1 — a  result  which  is  clearly 
impossible.  Hence  we  must  in  any  case  reach  a  contradiction 
of  the  hypothesis.     Thus  the  theorem  is  proved. 

The  preceding  results  are  special  cases  of  the  following  more 
general  theorem : 


ELEMENTARY   PROPERTIES    OF   INTEGERS  13 

III.  Among  the  integers  of  the  arithmetic  progression  a,  a-\-d, 
a-\-2d,  a-\-$d3  .  .  .  ,  there  is  an  infinite  number  of  primes,  pro- 
vided that  a  and  b  are  relatively  prime. 

For  the  special  case  given  in  theorem  II  we  have  an  elemen- 
tary proof;  but  for  the  general  theorem  the  proof  is  difficult. 
We  shall  not  give  it  here. 

EXERCISES 

i.  Prove  that  there  is  an  infinite  number  of  primes  of  the  form  4»  — i.     , 

2.  Show  that  an  odd  prime  number  can  be  represented  as  the  difference  of 
two  squares  in  one  and  in  only  one  way. 

3.  The  expression  mp  —  np,  in  which  m  and  n  are  integers  and  p  is  a  prime, 
is  either  prime  to  p  or  is  divisible  by  p2. 

4.  Prove  that  any  prime  number  except  2  and  3  is  of  one  of  the  forms  6w+i, 
6n—  1. 

§  5.    The  Fundamental  Theorem  of  Euclid 

//  a  and  b  are  any  two  positive  integers  there  exist  integers 
q  and  r,  q>o,  o=r<b,  such  that 

a  =  qb-\-r.. 

If  a  is  a  multiple  of  b  the  theorem  is  at  once  verified,  r  being 
in  this  case  o.  If  a  is  not  a  multiple  of  b  it  must  lie  between 
two  consecutive  multiples  of  b;  that  is,  there  exists  a  q  such 
that 

qb<a<(q+i)b. 

Hence  there  is  an  integer  r,  o<r<b,  such  that  a  =  qb-\-r.  In 
case  b  is  greater  than  a  it  is  evident  that  q  =  o  and  r  =  a.  Thus 
the  proof  of  the  theorem  is  complete. 

§  6.  Divisibility  by  a  Prime  Number 

I.  If  p  is  a  prime  number  and  m  is  any  integer,  then  m  either 
is  divisible  by  p  or  is  prime  to  p. 

This  theorem  follows  at  once  from  the  fact  that  the  only 
divisors  of  p  are  i  and  p. 

II.  The  product  of  two  integers  each  less  than  a  given  prime 
number  p  is  not  divisible  by  p. 


14  THEORY   OF   NUMBERS 

Let  a  be  a  number  which  is  less  than  p  and  suppose  that  b 
is  a  number  less  than  p  such  that  ab  is  divisible  by  p,  and  let 
b  be  the  least  number  for  which  ab  is  so  divisible.  Evidently 
there  exists  an  integer  m  such  that 

mb<p<(m  +  i)b. 

Then  p—mb<b.  Since  ab  is  divisible  by  p  it  is  clear  that  mab 
is  divisible  by  p;  so  is  ap  also;  and  hence  their  differei^B 
ap  —  mab,  =a(p  —  mb),  is  divisible  by  p.  That  is,  the  prodiH 
of  a  by  an  integer  less  than  b  is  divisible  by  p,  contrary  to  tH 
assumption  that  &  is  the  least  integer  such  that  ab  is  divisibH 
by  p.  The  assumption  that  the  theorem  is  not  true  has  thiH 
led  to  a  contradiction;   and  thus  the  theorem  is  proved. 

III.  If  neither  of  two  integers  is  divisible  by  a  given  primm 
number  p  their  product  is  not  divisible  by  p. 

Let  a  and  b  be  two  integers  neither  of  which  is  divisible 
by  the  prime  p.  According  to  the  fundamental  theorem  off 
Euclid  there  exist  integers  m,  n,  a,  /3  such  that 

a  =  mp-\-a,    o<a<p} 
b  =  np+(3,     o<(3<p. 
Then  ab  =  (mp+a)(np+p)  =  (mnp+<*+p)p+aP. 

If  now  we  suppose  ab  to  be  divisible  by  p  we  have  a/3  divisible 
by  p.  This  contradicts  II,  since  a  and  /3  are  less  than  p.  Hence 
ab  is  not  divisible  by  p. 

By  an  application  of  this  theorem  to  the  continued  product, 
of  several  factors,  the  following  result  is  readily  obtained: 

IV.  If  no  one  of  several  integers  is  divisible  by  a  given  prime ! 
p  their  product  is  not  divisible  by  p. 

§  7.  The  Unique  Factorization  Theorem 

I.  Every  integer  greater  than  unity  can  be  represented  in  one 
and  in  only  one  way  as  a  product  of  prime  num1  ers. 

In  the  first  place  we  shall  show  that  it  "s  a T ways  possible 
to  resolve  a  given  integer  m  greater  than  v  lty  into  prime 


ELEMENTARY   PROPERTIES   OF   INTEGERS  15 

factors  by  a  finite  number  of  operations.  In  the  proof  of  the- 
orem I,  §  3,  we  showed  how  to  find  a  prime  factor  pi  of  m  by 
a  finite  number  of  operations.     Let  us  write 

m  =  p\m\. 

If  mi  is  not  unity  we  may  now  find  a  prime  factor  p2  of  nil. 
Then  we  may  write 

m  =  pinii  =  pip2ffi2- 

If  ni2  is  not  unity  we  may  apply  to  it  the  same  process  as  that 
applied  to  nil  and  thus  obtain  a  third  prime  factor  of  m.  Since 
mi>m2>ni3>  ...  it  is  clear  that  after  a  finite  number  of 
operations  we  shall  arrive  at  a  decomposition  of  m  into  prime 
factors.     Thus  we  shall  have 

m  =  pip2  •  .  .  pr 

where  pi,  p2,  .  .  .  ,  pr  are  prime  numbers.  We  have  thus 
proved  the  first  part  of  our  theorem,  which  says  that  the  decom- 
position of  an  integer  (greater  than  unity)  into  prime  factors 
is  always  possible. 

Let  us  now  suppose  that  we  have  also  a  decomposition  of 
m  into  prime  factors  as  follows: 

m  =  qiq2  .   .   .  qs. 
Then  we  have 

pip2  •  •  .  pr  =  qiq2  .  .  .  q* 

Now  pi  divides  the  first  member  of  this  equation.  Hence  it 
also  divides  the  second  member  of  the  equation.  But  pi  is 
prime;  and  therefore  by  theorem  IV  of  the  preceding  section 
we  see  that  pi  divides  some  one  of  the  factors  q\  we  suppose 
that  pi  is  a  factor  of  qi.  It  must  then  be  equal  to  qi.  Hence 
we  have 

p2p3  •  •  -  pr  =  q2q3  .  .  .  qa. 

By  the  same  argument  we  prove  that  p2  is  equal  to  some  q, 
say  qo.     Then  we  have 

p3p±  •  •  •  pr  =  qzq±  .  .  .  q„ 


16  THEORY   OF   NUMBERS 

Evidently  the  process  may  be  continued  until  one  side  of  the 
equation  is  reduced  to  i.  The  other  side  must  also  be  reduced 
to  i  at  the  same  time.  Hence  it  follows  that  the  two  decom- 
positions of  m  are  in  fact  identical. 

This  completes  the  proof  of  the  theorem. 

The  result  which  we  have  thus  demonstrated  is  easily  the 
most  important  theorem  in  the  theory  of  integers.  It  can 
also  be  stated  in  a  different  form  more  convenient  for  some 
purposes: 

II.  Every  non-unit  positive  integer  m  can  be  represented  in 
one  and  in  only  one  way  in  the  form 


m  =  p1aip2a2  .  .  .  pt 


an 


where  pi,  p2,  .  .  .  ,  pn  are  different  primes  and  «i,  qe,  .  •  •  , 
un  are  positive  integers. 

This  comes  immediately  from  the  preceding  representation 
of  m  in  the  form  m  =  pip2  .  .  .  pr  by  combining  into  a  power 
of  pi  all  the  primes  which  are  equal  to  pi. 

COROLLARY  i.  If  a  and  b  are  relatively  prime  integers 
and  c  is  divisible  by  both  a  and  b,  then  c  is  divisible  by  ab. 

COROLLARY  2.  If  a  and  b  are  each  prime  to  c  then  ab  is 
prime  to  c. 

Corollary  3.  If '  a  is  prime  to  c  and  ab  is  divisible  by  c, 
then  b  is  divisible  by  c. 

§  8.  The  Divisors  of  an  Integer 

The  following   theorem  is  an  immediate   corollary  of   the 
results  in  the  preceding  section : 
I.  All  the  divisors  of  m, 

are  of  the  form 

pi(ilp2fi2  .  .  .  pnPn,     o<ft^a*; 

and  every  such  number  is  a  divisor  of  m. 


/ 


ELEMENTARY   PROPERTIES   OF   INTEGERS  17 

From  this  it  is  clear  that  every  divisor  of  m  is  included  once 
and  only  once  among  the  terms  of  the  product 

(l+^l+/>l2+    •    •    •   +^ia0(l+/>2+^22+     •    •    .     +^2a>)     •    •    • 

(l+pn  +  pn2+    ...     +pnan), 

when  this  product  is  expanded  by  multiplication.  It  is  obvious 
that  the  number  of  terms  in  the  expansion  is  (ai  +  i)(«2  +  i)  •  •  • 
(a»+i).     Hence  we  have  the  theorem: 

II .  The  number  of  divisors  of  m  is  (a i  + 1 )  («2  + 1 )  •  •  .  («» + 1 )  • 
Again  we  have 

n(i+pi+pi2+  .  .  .  +y)«n^.1>"1. 

»  »     pi  —  i 

Hence, 

III.  The  sum  of  the  divisors  of  m  is 


piai+l-I     p2a*+\-I 


«n  +  l  _ 


pi -I  p2  —  l  pn—I 

In  a  similar  manner  we  may  prove  the  following  theorem: 
IV.  The  sum  of  the  hth  powers  of  the  divisors  of  m  is 

Pih-i    '  '  ' :  ' r  pnh-i    ' 

EXERCISES 

i.  Find  numbers  x  such  that  the  sum  of  the  divisors  of  x  is  a  perfect  square. 

2.  Show  that  the  sum  of  the  divisors  of  each  of  the  following  integers  is  twice 
the  integer  itself:  6,  28,  496,  8128,  33550336.  Find  other  integers  x  such  that 
the  sum  of  the  divisors  of  x  is  a  multiple  of  x. 

3.  Prove  that  the  sum  of  two  odd  squares  cannot  be  a  square." 
^4.  Prove  that  the  cube  of  any  integer  is  the  difference  of  the  squares  of  two 

integers. 
""> kg.  In  order  that  a  number  shall  be  the  sum  of  consecutive  integers,  it  is  neces- 
sary and  sufficient  that  it  shall  not  be  a  power  of  2. 

6.  Show  that  there  exist  no  integers  x  and  y  (zero  excluded)  such  that  y2  =  2x2. 
Hence,  show  that  there  does  not  exist  a  rational  fraction  whose  square  is  2. 

7.  The  number  m  =  pia^pta-  .  .  .  pnan,  where  the  p's  are  different  primes  and 
the  a's  are  positive  integers,  may  be  separated  into  relatively  prime  factors  in 
2s-1  different  ways. 

8.  The  product  of  the  divisors  of  m  is  \/mc  where  v  is  the  number  of  divisors 
of  m. 


18  THEORY   OF   NUMBERS 


§  9.  The  Greatest  Common  Factor  of  Two  or  More 
Integers 

Let  m  and  n  be  two  positive  integers  such  that  m  is  greater 
than  n.  Then,  according  to  the  fundamental  theorem  of 
Euclid,  we  can  form  the  set  of  equations 

m  =  qn-{-ni,  o<n\<n, 

n  =  qini-\-n2,  o<n2<ni, 

ni=q2n2+m,  o<n-s<n2, 


nk-2  =  qk-ink-i+nk,       o<nk<nk-i, 
nk-i=qknk.    +    0 

If  m  is  a  multiple  of  n  we  write  n  =  n0,  k  =  o,  in  the  above  equa- 
tions. 

DEFINITION.  The  process  of  reckoning  involved  in 
determining  the  above  set  of  equations  is  called  the  Euclidian 
Algorithm. 

I.  The  number  nk  to  which  the  Euclidian  algorithm  leads  is 
the  greatest  common  divisor  of  m  and  n. 

In  order  to  prove  this  theorem  we  have  to  show  two  things  : 

1)  That  nk  is  a  divisor  of  both  m  and  n; 

2)  That  the  greatest  common  divisor  d  of  m  and  n  is  a 
divisor  of  nk. 

To  prove  the  first  statement  we  examine  the  above  set  of 
equations,  working  from  the  last  to  the  first.  From  the  last 
equation  we  see  that  nk  is  a  divisor  of  nk-\.  Using  this  result 
we  see  that  the  second  member  of  next  to  the  last  equation  is 
divisible  by  nk.  Hence  its  first  member  nk-2  must  be  divisible 
by  nk.  Proceeding  in  this  way  step  by  step  we  show  that 
U2  and  ni,  and  finally  that  n  and  m,  are  divisible  by  nk. 

For  the  second  part  of  the  proof  we  employ  the  same  set  of 
equations  and  work  from  the  first  one  to  the  last  one.  Let 
d  be  any  common  divisor  of  m  and  n.  From  the  first  equation 
we  see  that  d  is  a  divisor  of  ni.  Then  from  the  second  equation 
it  follows  that  J  is  a  divisor  of  n2.     Proceeding  in  this  way  we 


ELEMENTARY   PROPERTIES    OF   INTEGERS  19 

show  finally  that  d  is  a  divisor  of  fit.  Hence  any  common 
divisor,  and  in  particular  the  greatest  common  divisor,  of  m 
and  n  is  a  factor  of  nk. 

This  completes  the  proof  of  the  theorem. 

Corollary.  Every  common  divisor  of  m  and  n  is  a  factor 
of  their  greatest  common  divisor. 

II.  Any  number  n%  in  the  above  set  of  equations  is  the  differ- 
ence of  multiples  of  m  and  n. 

From  the  first  equation  we  have 

n\=m  —  qn 

so  that  the  theorem  is  true  for  i  =  i.  We  shall  suppose  that 
the  theorem  is  true  for  every  subscript  up  to  i—i  and  prove 
it  true  for  the  subscript  i.     Thus  by  hypothesis  we  have  * 

nt-2  =  ±(a*-2W  — j8<-2«), 

ni-i  =  zF(ai-im  —  (3i-in). 
Substituting  in  the  equation 

we  have  a  result  of  the  form 

Hi  =  zb  (at  m  —  fan) . 

From  this  we  conclude  at  once  to  the  truth  of  the  theorem. 

Since  nt  is  the  greatest  common  divisor  of  m  and  n,  we  have 
as  a  corollary  the  following  important  theorem: 

III.  //  d  is  the  greatest  common  divisor,  of  the  positive  integers 
m  and  n,  then  there  exist  positive  integers  ajindJLsuch  that 

am—ph=±d. 

If  we  consider  the  particular  case  in  which  m  and  n  are  rela- 
tively prime,  so  that  d^h  we  see  that  there  exist  positive 
integers  a  and  0  such  that  am  —  f}n=zLi.  Obviously,  if  m  and 
n  have  a  common  divisor  d,  greater  than  i,  there  do  not  exist 

*  If  i  =  2  we  must  replace  w^_2  by  n. 


20 


THEORY  OF   NUMBERS 


integers  a  and  /3  satisfying  this  relation;  for,  if  so,  d  would  be 
a  divisor  of  the  first  member  of  the  equation  and  not  of  the 
second.     Thus  we  have  the  following  theorem: 

IV.  A  necessary  and  sufficient  condition  that  m  and  n  are 
relatively  prime  is  that  there  exist  integers  a  and  /3  such  that 
am—(3n  =  zki. 

The  theory  of  the  greatest  common  divisor  of  three  or  more 
numbers  is  based  directly  on  that  of  the  greatest  common 
divisor  of  two  numbers;  consequently  it  does  not  require  to 
be  developed  in  detail. 

EXERCISES 

i.  If  d  is  the  greatest  common  divisor  of  m  and  n,  then  m/d  and  n/d  are  rela- 
tively prime. 

2.  If  <i  is  the  greatest  common  divisor  of  m  and  n  and  k  is  prime  to  n,  then 
d  is  the  greatest  common  divisor  of  km  and  n. 

3.  The  number  of  multiplies  of  b  in  the  sequence  a,  2a,  3a,  .  .  .  ,  ba  is  equal 
to  the  greatest  common  divisor  of  a  and  b. 

~p?-  4.  If  the  sum  or  the  difference  of  two  irreducible  fractions  is  an  integer,  the 
denominators  of  the  fractions  are  equal. 

5.  The  algebraic  sum  of  any  number  of  irreducible  fractions,  whose  denomi- 
nators are  prime  each  to  each,  cannot  be  an  integer. 

6*.  The  number  of  divisions  to  be  effected  in  finding  the  greatest  common 
divisor  of  two  numbers  by  the  Euclidian  algorithm  does  not  exceed  five  times 
the  number  of  digits  in  the  smaller  number  (when  this  number  is  written  in  the 
usual  scale  of  10). 

§  10.  The  Least  Common  Multiple  of  Two  or  More 

Integers 

I.  The  common  multiples  of  two  or  more  numbers  are  the 
multiples  of  their  least  common  multiple. 

This  may  be  readily  proved  by  means  of  the  unique  factori- 
zation theorem.  The  method  is  obvious.  We  shall,  however, 
give  a  proof  independent  of  this  theorem. 

Consider  first  the  case  of  two  numbers;  denote  them  by 
m  and  n  and  their  greatest  common  divisor  by  d.  Then  we 
have 

m  =  dfx,     n  =  dv, 

where  fi  and  v  are  relatively  prime  integers.     The  common 
multiples  sought  are  multiples  of  m  and  are  all  comprised  in  the 


ELEMENTARY   PROPERTIES   OF   INTEGERS  .  21 

numbers  am,  =  adfx,  where  a  is  any  integer  whatever.  In  order 
that  these  numbers  shall  be  multiples  of  n  it  is  necessary  and 
sufficient  that  ad/x  shall  be  a  multiple  of  dv;  that  is,  that  a/x 
shall  be  a  multiple  of  v;  that  is,  that  a  shall  be  a  multiple  of 
v,  since  /*  and  v  are  relatively  prime.  Writing  a  =  bv  we  have 
as  the  multiples  in  question  the  set  bdjiv  where  5  is  an  arbitrary 
integer.  This  proves  the  theorem  for  the  case  of  two  numbers; 
for  dfiv  is  evidently  the  least  common  multiple  of  m  and  n. 

We  shall  now  extend  the  proposition  to  any  number  of 
integers  m,  n,  p,  q,  .  .  .  .  The  multiples  in  question  must 
be  common  multiples  of  m  and  n  and  hence  of  their  least  common 
multiple  p.  Then  the  multiples  must  be  multiples  of  ju  and  p 
and  hence  of  their  least  common  multiple  pi.  But  m  is  evi- 
dently the  least  common  multiple  of  m,  n,  p.  Continuing  in  a 
similar  manner  we  may  show  that  every  multiple  in  question 
is  a  multiple  of  /i,  the  least  common  multiple  of  m,  n,  p,  q,  .  .  .  . 
And  evidently  every  such  number  is  a  multiple  of  each  of  the 
numbers  m,  n,  p,  q,  .  .  .  . 

Thus  the  proof  of  the  theorem  is  complete. 

When  the  two  integers  m  and  n  are  relatively  prime  their 
greatest  common  divisor  is  i  and  their  least  common  multiple 
is  their  product.  Again  if  p  is  prime  to  both  m  and  n  it  is  prime 
to  their  product  mn\  and  hence  the  least  common  multiple 
of  m}  n,  p  is  in  this  case  mnp.  Continuing  in  a  similar  manner 
we  have  the  theorem : 

II.  The  least  common  multiple  of  several  integers,  prime 
each  to  each,  is  equal  to  their  product. 

EXERCISES 

i.  In  order  that  a  common  multiple  of  n  numbers  shall  be  the  least,  it  is  neces- 
sary and  sufficient  that  the  quotients  obtained  by  dividing  it  successively  by  the 
numbers  shall  be  relatively  prime. 

2.  The  product  of  n  numbers  is  equal  to  the  product  of  their  least  common 
multiple  by  the  greatest  common  divisor  of  their  products  n  —  i  at  a  time. 

3.  The  least  common  multiple  of  n  numbers  is  equal  to  any  common  mul- 
tiple M  divided  by  the  greatest  common  divisor  of  the  quotients  obtained  on 
dividing  this  common  multiple  by  each  of  the  numbers. 

4.  The  product  of  n  numbers  is  equal  to  the  product  of  their  greatest  common 
divisor  by  the  least  common  multiple  of  the  products  of  the  numbers  taken  n—  1 
at  a  time. 


22  THEORY   OF   NUMBERS 


§  ii.  Scales  of  Notation 

I.  If  m  and  n  are  positive  integers  and  n>i,  then  m  can  be 
represented  in  terms  of  n  in  one  and  in  only  one  way  in  the  form 

ni  =  a0nh+a1n?>~1-\-  .  .  .   +aA_1w+aftj 
where 

ao^o,     o^ai<n,     i  =  o,  i,  2,  .  .  .  ,  h. 

That  such  a  representation  of  m  exists  is  readily  proved  by 
means  of  the  fundamental  theorem  of  Euclid.     For  we  have 

.  ni  =  non+ah,  o^ah<n, 

n0  =  nin+ah-u  o^ah-i<n, 


ni  =  n2n-\-ah-2,  o^ah-2<n} 

nhs  =nh-2n+a2,  o^a2<n, 

nh-2   =nh-in+ai,  o^ci<n, 

nh-i=ao,  o<ao<n. 

If  the  value  of  nh-i  given  in  the  last  of  these  equations  is  sub- 
stituted in  the  second  last  we  have 

nh-2  =  aon+ai. 

This  with  the  preceding  gives 

nh-3  =  aon2+ain+a2. 

Substituting  from  this  in  the  preceding  and  continuing  the 
process  we  have  finally 

m  =  aon7l+ainn~l-\-  .  .  .  +ah_1n+ah, 

a  representation  of  m  in  the  form  specified  in  the  theorem. 

To  prove  that  this  representation  is  unique,  we  shall  suppose 
that  m  has  the  representation 

ni  =  b0nk+b1nk~1  +  .  .  .  -^b^-^n-^h, 
where 

frop^o,     o<bi<n,     i  =  o,  1,  2,  .  .  .  ,  k, 


ELEMENTARY   PROPERTIES    OF   INTEGERS  23 

and  show  that  the  two  representations  are  identical.  We 
■have 

a0nh  +  .  .  .  +ah-in+ah  =  b0nk-+  .  .  .   +h-in+bk. 

Then 

aonh+  .  .  .   +ah-in-(b0nk+  .  .  .   +bk-iti)=bk  —  ah. 

The  first  member  is  divisible  by  n.  Hence  the  second  is  also. 
But  the  second  member  is  less  than  n  in  absolute  value;  and 
hence,  in  order  to  be  divisible  by  n,  it  must  be  zero.  That  is, 
bt  =  ah.  Dividing  the  equation  through  by  n  and  transposing 
we  have 

a0nh~1+  .  .  .   -\-ah-2n  —  (b0nk-l-{-  .  .  .   +bk-2n)=bk-i—ah-i 

It  may  now  be  seen  that  bk~i=afl-i.  It  is  evident  that  this 
process  may  be  continued  until  either  the  a's  are  all  eliminated 
from  the  equation  or  the  b's  are  all  eliminated.  But  it  is 
obvious  that  when  one  of  these  sets  is  eliminated  the  other  is 
also.  Hence,  h  =  k.  Also,  every  a  equals  the  b  which  multi- 
plies the  same  power  of  n  as  the  corresponding  a.  That  is, 
the  two  representations  of  m  are  identical.  Hence  the  repre- 
sentation in  the  theorem  is  unique. 

From  this  theorem  it  follows  as  a  special  case  that  any  posi- 
tive integer  can  be  represented  in  one  and  in  only  one  way  in 
the  scale  of  10;  that  is,  in  the  familiar  Hindoo  notation.  It 
can  also  be  represented  in  one  aild  in  only  one  way  in  any  other 
scale.     Thus 

120759  =  i. 76+o.75  +  i.74  +  2.73+o.72+3.71  +  2. 

Or,  using  a  subscript  to  denote  the  scale  of  notation,  this  may 
be  written 

(120759)10  =  (1012032)7. 

For  the  case  in  which  n  (of  theorem  I)  is  equal  to  2,  the 
only  possible  values  for  the  a's  are  o  and  1.  Hence  we  have 
at  once  the  following  theorem: 

II.  Any  positive  integer  can  be  represented  in  one  and  in  only 
one  way  as  a  sum  of  different  powers  of  2. 


24 


THEORY   OF   NUMBERS 


EXERCISES 

i.  Any  positive  integer  can  be  represented  as  an  aggregate  of  different  powers 
of  3,  the  terms  in  the  aggregate  being  combined  by  the  signs  +  and  —  appropri- 
ately chosen. 

2.  Let  m  and  n  be  two  positive  integers  of  which  n  is  the  smaller  and  suppose 
that  2  Sn<2  +  .  By  means  of  the  representation  of  m  and  n  in  the  scale  of 
2  prove  that  the  number  of  divisions  to  be  effected  in  finding  the  greatest  common 
divisor  of  m  and  »  by  the  Euclidian  algorithm  does  not  exceed  2k. 


§  12.  Highest  Power  of  a  Prime  p  Contained  in  n\. 

Let  n  be  any  positive  integer  and  p  any  prime  number  not 
greater  than  n.  We  inquire  as  to  what  is  the  highest  power 
p^oi  the  prime  p  contained  in  n\. 

In  solving  this  problem  we  shall  find  it  convenient  to  employ 
the  notation 


[7] 


to  denote  the   greatest  integer   a  such  that  as^r.     With  this 
notation  it  is  evident  that  we  have 


and  more  generally 


M 

L  p1 


(1) 


If  now  we  use  H  { x }  to  denote  the  index  of  the  highest  power 
of  p  contained  in  an  integer  x,  it  is  clear  that  we  have 


(0 


since  only  multiples  of  p  contain  the  factor  p.     Hence 

sl.„-[l]+s{..,...g]}. 


ELEMENTARY  PROPERTIES   OF   INTEGERS  25 

Applying  the  .same  process  to  the  H-i unction  in  the  second 
member  and  remembering  relation  (i)  it  is  easy  to  see  that 
we  have 

Continuing  the  process  we  have  finally 

Bi«H3+K+r?]+ •  •  •  ■ 

the  series  on  the  right  containing  evidently  only  a  finite  num- 
ber of  terms  different  from  zero.     Thus  we  have  the  theorem: 

I.   The  index  of  the  highest  power  of  a  prime  p  contained 
in  nl  is 


RMfM?] 


+  .  .  . 


The  theorem  just  obtained  may  be  written  in  a  different 
form,  more  convenient  for  certain  of  its  applications.  Let 
n  be  expressed  in  the  scale  of  p  in  the  form 

n  =  a0ph+a1ph~l  +  .  .  .   -\-ah-!p-\-ah, 
where 

aoT^o,     o^at<p,     i  =  o,  i,  2,  .  .  .  ,  h. 

Then  evidently 

[^]=ao/>*-1.+ai/>*-2+  •  •  •   +a*-2p+ah-1, 


26  THEORY   OF   NUMBERS 

Adding  these  equations  member  by  member  and  combining 
the  second  members  in  columns  as  written,  we  have 


m?m 


+ . 


i-o       p—i 

^aoph+aip?t~1+  .  .  .   +ah-(ao+ai+    .  .  .   +ah) 

p-i 
=  n  —  (ao+ai  +  .  .  .   +flft) 
p-i 

Comparing  this  result  with  theorem  I  we  have  the  following 
theorem : 

II.  If  n  is  represented  in  the  scale  of  p  in  the  form 

n  =  a0ph+aiph-1+  .  .  .   +ah, 

where  p  is  prime  and 

a05*o,     o^al<p,     i  =  o,  i,  2,  .  .  .,  h, 

then  the  index  of  the  highest  power  of  p  contained  in  n !  is 

n  —  (ao+ai-\-  .  .  .  +ah) 
p-i 

Note  the  simple  form  of  the  theorem  for  the  case  p  =  2; 
in  this  case  the  denominator  p  —  i  is  unity. 

We  shall  make  a  single  application  of  these  theorems  by 
proving  the  following  theorem: 

III.  If  n,  a,  (3,  ...  j  \  are  any  positive  integers  such  that 

n=a+fi+  .  .  .   +X,  then 

^TTTTT!  {A) 

is  an  integer. 

Let  p  be  any  prime  factor  of  the  denominator  of  the  frac- 
tion (A).  To  prove  the  theorem  it  is  sufficient  to  show  that 
the  index  of  the  highest  power  of  p  contained  in  the  numerator 
is  at  least  as  great  as  the  index  of  the  highest  power  of  p  con- 


ELEMENTARY   PROPERTIES   OF   INTEGERS 


27 


tained  in  the  denominator.     This  index  for  the  denominator 
is  the  sum  of  the  expressions 


[f]+[?l+[? 

mm? 


+  .  .  . 


+  . 


The  corresponding  index  for  the  numerator  is 


(B) 


m?\ 


+ 


+ 


(O 


But,  since  n=a+fi+  .  .  .   +  \,  it  is  evident  that 

Prom  this  and  the  expressions  in  (B)  and  (C)  it  follows  that 
the  index  of  the  highest  power  of  any  prime  p  in  the  numerator 
of  (A)  is  equal  to  or  greater  than  the  index  of  the  highest  power 
of  p  co'ntained  in  its  denominator.  The  theorem  now  follows 
at  once. 

Corollary.     The  product  of  n  consecutive  integers  is  divisible 
by  n\. 

EXERCISES 

i.  Show  that  the  highest  power  of  2  contained  in  1000!  is  2994;  in  1900!  is  21893. 
Show  that  the  highest  power  of  7  contained  in  10000!  is  71665. 

2.  Find  the  highest  power  of  72  contained  in  1000! 

3.  Show  that  1000!  ends  with  240  zeros. 

4.  Show  that  there  is  no  number  n  such  that  3'  is  the  highest  power  of  3  con- 
tained in  n\. 

5.  Find  the  smallest  number  n  such  that  the  highest  power  of  5  contained 
in  n\  is  531.     What  other  numbers  have  the  same  property? 

6.  If  n  =  rs,  r  and  s  being  positive   integers,  show  that  n\  is  divisible  by  (r!)'9; 
by  <>!)r;    by  the  least  common  multiple  of  (r!)5  and  (s\)T. 

7.  If  n^a+p+pq+rs,  where  a,  /3,  p,  q,  r,  s,  are  positive  integers,  then  n\  is 
divisible  by 


28  THEORY   OF   NUMBERS 

S.  When  m  and  n  are  two  relatively  prime  positive  integers  the  quotient 

(m+n  —  i)! 

mini 
as  an  integer. 

9*.  If  m  and  n  are  positive  integers,  then  each  of  the  quotients 

(mn)l  (2w)!(2»)! 

w!(w!)n'  mlnl(m-\-n)l! 

is  an  integer.     Generalize  to  &  integers  w,  n,  p,  .  .  .  . 

10*.     If  »=a+/3+^g+rs  where  a,  /3,  p,  q,  r,  s  are  positive  integers,  then  n\ 
is  divisible  by 

ali3lrlpl(ql)p(sl)r. 
ii*.  Show  that 

(rst)l        / 

is  an  integer  (r,  s,  t  being  positive  integers).     Generalize  to  the  case  of  n  integers 
r,  s,  t,u,  .  .  .  . 


§  13.  Remarks  Concerning  Prime  Numbers 

We  have  seen  that  the  number  of  primes  is  infinite.  But 
the  integers  which  have  actually  been  identified  as  prime  are 
finite  in  number.  Moreover,  the  question  as  to  whether  a  large 
number,  as  for  instance  2257  —  1,  is  prime  is  in  general  very 
difficult  to  answer.  Among  the  large  primes  actually  identified 
as  such  are  the  following: 


,61 


I,       275-5  +  I,       289-I,       2127-I, 


No  analytical  expression  for  the  representation  of  prime  num- 
bers has  yet  been  discovered.  Fermat  believed,  though  he  con- 
fessed that  he  was  unable  to  prove,  that  he  had  found  such  an 
analytical  expression  in 

22    +1. 

Euler  showed  the  error  of  this  opinion  by  finding  that  641  is  a 
factor  of  this  number  for  the  case  when  ^  =  5. 

The  subject  of  prime  numbers  is  in  general  one  of  exceeding 
difficulty.     In  fact  it  is  an  easy  matter  to  propose  problems 


ELEMENTARY   PROPERTIES   OF   INTEGERS  29 

about  prime  numbers  which  no  one  has  been  able  to  solve. 
Some  of  the  simplest  of  these  are  the  following: 

i.  Is  there  an  infinite  number  of  pairs  of  primes  differing 
by  2? 

2.  Is  every  even  number  (other  than  2)   the  sum  of  two 
primes  or  the  sum  of  a  prime  and  the  unit? 

3.  Is  every  even  number  the  difference  of  two  primes  or 
the  difference  of  1  and  a  prime  number? 

4.  To  find  a  prime  number  greater  than  a  given  prime. 

5.  To  find  the  prime  number  which  follows  a  given  prime. 
jo\/To  find  the  number  of  primes  not  greater  than  a  given 

number. 

7.  To  compute  directly  the  nth  prime  number,  when  n  is 
given. 


CHAPTER  II 

ON  THE  INDICATOR  OF  AN  INTEGER 

§  14.  Definition.     Indicator  of  a  Prime  Power 

Definition.  If  m  is  any  given  positive  integer  the  num- 
ber of  positive  integers  not  greater  than  m  and  prime  to  it  is 
called  the  indicator  of  m.  It  is  usually  denoted  by  <j>(m),  and 
is  sometimes  called  Euler's  ^-function  of  m.  More  rarely, 
it  has  been  given  the  name  of  totient  of  m. 

As  examples  we  have 

4>(i)  =  1,     0(2)  =  1,     0(3)  =  2,     0(4)  =2. 
If  p  is  a  prime  number  it  is  obvious  that 

for  each  of  the  integers  1,2,3,  .  .  .  ,  p  —  1  is  prime  to  p. 

Instead  of  taking  m  =  p  let  us  assume  that  m  =  pa,  where 
a  is  a  positive  integer,  and  seek  the  value  of  4>{pa).  Obviously, 
every  number  of  the  set  1,  2,  3,  ...  ,  ^either  is  divisible 
by  p  or  is  prime  to  p\  The  number  of  integers  in  the  set 
divisible  by  p  is  p"'1.  Hence  pa  —  pa~1  of  them  are  prime 
to  p.     Hence  4>{pa)  =  pa-pa~1.     Therefore 

If  p  is  any  prime  number  and  a  is  any  positive  integer,  then 


*<r)=r(i-|-} 


§  15.  The  Indicator  of  a  Product 

I.  If  (i  and  v  are  any  two  relatively  prime  positive  integers 
then 

<f>(fj.v)  =<£(m)0(V). 

•80 


ON   THE   INDICATOR   OF   AN   INTEGER  31 

In  order  to  prove  this  theorem  let  us  write  all  the  integers 
up  to  iiv  in  a  rectangular  array  as  follows: 

i  2  3  .  .  .  h  .  .  .    m 

jliH-I  jU+2  ^+3   .   .   .  M-f^  .   .   .  2/x 

2/i  +  I  2jU  +  2  2/Z+3    .   .   .  2fi+k   .   .    .  3M 


(y-l)jti+I    (y-l)/*  +  2    (i>-i)m+3   •   •   •   (^-i)m+^   .   . 


V]± 


(A) 


If  a  number  /z  in  the  first  line  of  this  array  has  a  factor  in 
common  with  /x  then  every  number  in  the  same  column  with 
h  has  a  factor  in  common  with  /*.  On  the  other  hand  if  h  is 
prime  to  n  so  is  every  number  in  the  column  with  h  at  the  top. 
But  the  number  of  integers  in  the  first  row  prime  to  /x  is  <£(/*). 
Hence  the  number  of  columns  containing  integers  prime  to  fi 
is  0(/x)  and  every  integer  in  these  columns  is  prime  to  p. 

Let  us  now  consider  what  numbers  in  one  of  these  columns 
are  prime  to  v\  for  instance,  the  column  with  h  at  the  top. 
We  wish  to  determine  how  many  integers  of  the  set 

h,     n+h,     2>i+h,  .  .  .  ,     (p—i)n+h 

are  prime  to  v.    Write 

sn+h  =  qsv-\-r5 

where  5  ranges  over  the  numbers  s  =  o,  i,  2,  .  .  .  ,  v  —  i  and 
o£r,0.  Clearly  sn+h  is  or  is  not  prime  to  v  according  as 
r8  is  or  is  not  prime  to  v.  Our  problem  is  then  reduced  to  that 
of  determining  how  many  of  the  quantities  rs  are  prime  to  v. 

First  let  us  notice  that  all  the  numbers  rs  are  different; 
for,  if  rs  =  r%  then  from 

sn+h  =  q8v+rs,     tii+h  =  qtv+rt, 

we  have  by  subtraction  that  (s—t)n  is  divisible  by  v.  But 
fx  is  prime  to  v  and  s  and  /  are  each  less  than  v.  Hence  (s  —  t)fj, 
can  be  a  multiple  of  v  only  by  being  zero;  that  is,  s  must  equal  t. 
Hence  no  two  of  the  remainders  rs  can  be  equal. 

Now  the  remainders  rs  are  v  in  number,  are  all  zero  or  posi- 
tive, eat:h  is  less  than  v,  and  they  are  all  distinct.     Hence  they 


32  THEORY   OF   NUMBERS 

are  in  some  order  the  numbers  o,  i,  2,  .  .  .  ,  v  —  1.  The  num- 
ber of  integers  in  this  set  prime  to  v  is  evidently  4>{v). 

Hence  it  follows  that  in  any  column  of  the  array  (A)  in  which 
the  numbers  are  prime  to  fi  there  are  just  </>(*>)  numbers  which 
are  prime  to  v.  That  is,  in  this  column  there  are  just  4>(v) 
numbers  which  are  prime  to  iiv.  But  there  are  <£(/x)  such 
columns.  Hence  the  number  of  integers  in  the  array  (A) 
prime  to  fxv  is  0(m)  4>{v)  . 

But  from  the  definition  of  the  ^-function  it  follows  that 
the  number  of  integers  in  the  array  (A)  prime  to  ixv  is  4>(jiv). 
Hence, 

which  is  the  theorem  to  be  proved. 

Corollary.  In  the  series  of  n  consecutive  terms  of  an 
arithmetical  progression  the  common  difference  of  which  is  prime 
to  n,  the  number  of  terms  prime  to  n  is  <f>(n). 

From  theorem  I  we  have  readily  the  following  more  general 
result : 

II.  If  mi,  W2,  .  .  .  ,  mk  are  k  positive  integers  which  are 
prime  each  to  each,  then 

4>(mim2  .  .  .  mk)  =  <t>(mi) <j>(m2)  .  .  .   <i>(mk).. 

§  16.  The,  Indicator  of  any  Positive  Integer 

From  the  results  of  §§14  and  15  we  have  an  immediate 
proof  of  the  following  fundamental  theorem: 

If  m  =  p1aip2a2  .  .  .  pnan  where  pi,  p2,  .  .  .  ,  pn  are  differ- 
ent primes  and  a\,  a2,  .  .  •  ,  a„  are  positive  integers,  then 

m=m(i-£)(t-±^  .  .  .  (x-i). 

For, 

4>{m)  =  4>{p^)<t>(p2a*)  .  .  .  4>{pnan) 


ON   THE   INDICATOR   OF   AN   INTEGER  33 

On  account  of  the  great  importance  of  this  theorem  we  shall 
give  a  second  demonstration  of  it. 

It  is  clear  that  the  number  of  integers  less  than  m  and 
divisible  by  p\  is 

m 

The  number  of  integers  less  than  m  and  divisible  by  p2  is 

m 
}~2 

The  number  of  integers  less  than  m  and  divisible  by  pip2  is 

m 


p\p2 


Hence  the  number  of  integers  less  than  m  and  divisible  by 
either  p\  or  p2  is 


m 


Pl       p2       plp2 

Hence  the  number  of  integers  less  than  m  and  prime  to  p\p2  is 

m     m  .     m           I        i  \  /        i 
m — | =m[  i )    i 

pl       p2       Plp2  \  pl/  \  p2 

We  shall  now  show  that  if  the  number  of  integers  less  than 
m  and  prime  to  p\p2  .  .  .  pi,  where  i  is  less  than  n,  is 


X   pin     P2)  '  ' 


i--- 


then»  the  number  of  integers  less  than  m  and  prime  to  pip', 
.  .  •  pipt+i  is 


-)  •  •  •  («-—)• 

p2l  \  Pi+l/ 

From  this  our  theorem  will  follow  at  once  by  induction. 


i ii 

Pil\      P<< 


m—m\  i 


34  THEORY   OF   NUMBERS 

From  our  hypothesis  it  follows  that  the  number  of  integers 
less  than  m  and  divisible  by  at  least  one  of  the  primes 
pi,  P2,    .  .  •  ,  pi  is 

or 

P\  plp2  plp2pS 

where  the  summation  in  each  case  runs  over  all  numbers  of 
the  type  indicated,  the  subscripts  of  the  fs  being  equal  to  or 
less  than  i. 

Let  us  consider  the  integers  less  than  m  and  having  the 
factor  pi+i  but  not  having  any  of  the  factors  pi,  p2,  .  .  .  ,  pi. 
Their  number  is 


Pi+l       pi+1   (        P\  Plp2  Plp2p3 

where  the  summation  signs  have  the  same  significance  as  before. 
For  the  number  in  question  is  evidently  tn/pt+i  minus  the 
number  of  integers  not  greater  than  m/pt+i  and  divisible  by 
at  least  one  of  the  primes  pi,  p2,  .  .  .  ,  pu 

If  we  add  (A)  and  (B)  we  have  the  number  of  integers  less 
than  m  and  divisible  by  one  at  least  of  the  numbers  pi,  p2, 
.  .  .  ,  pi+i>    Hence  the  number  of  integers  less  than  m  and 
prime  to  pi,  p2,  .  .  .  ,  pi+i  is 

m—2j  — \-2j 2j 


'i         pip2         pip2pz 

where  now  in  the  summations  the  subscripts  run  from   i   to 
i+i.     This  number  is  clearly  equal  to 


w    I 


Pi)\        P2)    '  V      Pi+l)' 


From  this  result,  as  we  have  seen  above,  our  theorem  follows 
at  once  by  induction. 


ON   THE   INDICATOR   OF   AN   INTEGER  35 


§  17.  Sum  of  the  Indicators  of  the  Divisors  of  a  Number 

We  shall  first  prove  the  following  lemma: 

Lemma.  If  d  is  any  divisor  of  m  and  m  =  nd,  the  number 
of  integers  not  greater  than  m  which  have  with  m  the  greatest  com- 
mon divisor  d  is  <t>(n). 

Every  integer  not  greater  than  m  and  having  the  divisor 
d  is  contained  in  the  set  d,  2d,  3d,  .  .  .  ,  nd.  The  number  of 
these  integers  which  have  with  m  the  greatest  common  divisor 
d  is  evidently  the  same  as  the  number  of  integers  of  the  set 
1,  2,  .  .  .  ,  n  which  are  prime  to  m/d,  or  n;  for  ad  and  m  have 
or  have  not  the  greatest  common  divisor  d  according  as  a  is 
or  is  not  prime  to  m/d,  =n.  Hence  the  number  in  question 
is  cj>(n). 

From  this  lemma  follows  readily  the  proof  of  the  following 
theorem : 

If  di,  d2,  .  .  .  ,  dr  are  the  different  divisors  of  m,  then 

<t>(d1)  +  <f>(d2)+  .  .  .   +<f>(dr)=m. 

Let  us  define  integers  mi,  m2,  .  .  .  ,  mr  by  the  relations 
m  =  dimi=d2m2=   .  .  .    =dTmr. 

Now  consider  the  set  of  m  positive  integers  not  greater  than 
m,  and  classify  them  as  follows  into  r  classes.  Place  in  the 
first  class  those  integers  of  the  set  which  have  with  m  the  great- 
est common  divisor  mi;  -their  number  is  <t>(di),  as  may  be  seen 
from  the  lemma.  Place  in  the  second  class  those  integers 
of  the  set  which  have  with  m  the  greatest  common  divisor  m2\ 
their  number  is  4>(d2)-  Proceeding  in  this  way  throughout, 
we  place  finally  in  the  last  class  those  integers  of  the  set  which 
have  with  m  the  greatest  common  divisor  mr',  their  number 
is  4>{dr).  It  is  evident  that  every  integer  in  the  set  falls  into 
one  and  into  just  one  of  these  r  classes.  Hence  the  total  num- 
ber m  of  integers  in  the  set  is  4>{d\) -\- 4>{d2) -{-  .  .  .  -\-<j>(dr). 
From  this  the  theorem  follows  immediately. 


36  THEORY   OF   NUMBERS 

EXERCISES 

1.  Show  that  the  indicator  of  any  integer  greater  than  2  is  even. 

2.  Prove  that  the  number  of  irreducible  fractions  not  greater  than  1  and  with 
denominator  equal  to  n  is  <f>(n). 

3.  Prove  that  the  number  of  irreducible  fractions  not  greater  than  1  and 
with  denominators  not  greater  than  n  is 

*(i)+0(2)+*(3)+  •  -  .  +*(»). 

4.  Show  that  the  sum  of  the  integers  less  than  n  and  prime,  to  n  is  %n<f>{n) 
if  «>i. 

5.  Find  ten  values  of  x  such  that  <f>(x)  =  24. 

6.  Find  seventeen  values  of  *  such  that  </>(#)  =  72. 

7.  Find  three  values  of  n  for  which  there  is  no  x  satisfying  the  equation 

<f>(x)  =  271. 

8.  Show  that  if  the  equation 

<f)(x)  =  n 

has  one  solution  it  always  has  a  second  solution,  n  being  given  and  x  being  the 
unknown. 

9.  Prove  that  all  the  solutions  of  the  equation 

<f>(x)  =  4n—  2,    n>i, 
are  of  the  form  pa  and  2pa,  where  p  is  a  prime  of  the  form  45—1. 

10.  How  many  integers  prime  to  n  are  there  in  the  set 

a)   1-2,     2-3,     3-4,  .  .  .  ,     »(»+i)? 
V)   1-2-3,     2-3-4,     3-4-5,  .  .  .  ,     «(»+i)(»+2)? 
\   Hi      LI      Hi  »(»+i)? 


4) 


222  2 

1-2-3      2-3-4       3'4-5  »(»+i)(»+2). 


6.  6  6  6 

it*.  Find  a  method  for  determining  all  the  solutions  of  the  equation 

<j>(x)  =  n, 

where  n  is  given  and  x  is  to  be  sought. 

12*.  A  number  theory  function  <f>(n)  is  denned  for  every  positive  integer  n; 
and  for  every  such  number  n  it  satisfies  the  relation 

<t>{di)+<t>{d2)+  .  .  .  +<f>(dr)=n, 

where  di,  (k,  .  .  .  ,  dr  are  the  divisors  of  n.     From  this  property  alone  show 

that 

«"-(-i)(-s)--  (-£>■' 

where  Pi,  P2,  •  •  •  ,  />*  are  the  different  prime  factors  of  n. 


CHAPTER  III 
ELEMENTARY  PROPERTIES  OF  CONGRUENCES 

§  1 8.  Congruences  Modulo  m 

Definitions.  If  a  and  .b  are  any  two  integers,  positive 
or  zero  or  negative,  whose  difference  is  divisible  by  m,  a  and  b 
are  said  to  be  congruent  modulo  m,  or  congruent  for  the  modulus 
m,  or  congruent  according  to  the  modulus  m.  Each  of  the 
numbers  a  and  b  is  said  to  be  a  residue  of  the  other. 

To  express  the  relation  thus  defined  we  may  write 

a  =  b-\-cm, 

where  c  is  an  integer  (positive  or  zero  or  negative).  It  is  more 
convenient,  however,  to  use  a  special  notation  due  to  Gauss, 
and  to  write 

a=b  mod  m,  , 

an  expression  which  is  read  a  is  congruent  to  b  modulo  m,  or 
a  is  congruent  to  b  for  the  modulus  m,  or  a  is  congruent  to  b 
according  to  the  modulus  m.  This  notation  has  the  advantage 
that  it  involves  only  the  quantities  which  are  essential  to  the 
idea  involved,  whereas  in  the  preceding  expression  we' had  the 
irrelevant  integer  c.  The  Gaussian  notation  is  of  great  value 
and  convenience  in  the  study  of  the  theory  of  divisibility. 
In  the  present  chapter  we  develop  some  of  the  fundamental 
elementary  properties  of  congruences.  It  will  be  seen  that 
many  theorems  concerning  equations  are  likewise  true  of  con- 
gruences with  fixed  modulus;  and  ft  is  this  analogy  with  equa- 
tions which  gives  congruences  (as  such)  one  of  their  chief  claims 
to  attention.  * 

37 


38  THEORY   OF   NUMBERS 

As  immediate  consequences  of  our  definitions  we  have  the 
following  fundamental  theorems: 

I.  If  fl=cmodm,     Ncmodw, 

then  a=bmodm\ 

that  is,  for  a  given  modulus,  numbers  congruent  to  the  same  num- 
ber are  congruent  to  each  other. 

For,  by  hypothesis,  a  —  c=Cim,  b—c  =  C2m,  where  c\  and 
C2  are  integers.  Then  by  subtraction  we  have  a  —  b  =  (ci—C2)m\ 
whence  a  =  b  mod  m. 

II.  //  a  =  bmodm,    a=j8  mod  w, 
then  azka=b±(3  mod  m; 

that  is,  congruences  with  the  same  modulus  may  be  added  or  sub- 
tracted member  by  member. 

For,  by  hypothesis,  a  —  b  =  c\m,  a  —  ^  =  c2m;  whence 
(ad=a)  —  (bdtP)  =  (ci±c2)m.     Hence  a±a=^±/3modw. 

III.  //  a=b  mod  ot, 

then  ca=cb  mod  m, 

c  being  any  integer  whatever. 

The  proof  is  obvious  and  need  not  be  stated. 

IV.  If  a  =  bmodm,     a=(3modm, 

then  aa  m  b(3  mod  m ; 

that  is,  two  congruences  with  the  same  modulus  may  be  multiplied 
member  by  member. 

For,  we  have  a  =  b+c\m,  a  =  (3+c2m.  Multiplying  these  equa- 
tions member  by  member  we  have  aa  =  bp-\-m(bc2+ficiJrciC-2m). 
Hence  aa=b$  mod  m. 

A  repeated  use  of  this  theorem  gives  the  following  result: 

V.  If  a=b  mod  m, 

then  an  =  bn  mod  m 

whtre  n  is  any  positive  integer. 


ELEMENTARY   PROPERTIES   OF   CONGRUENCES  39 

As  a  corollary  of  theorems  II,  III  and  V  we  have  the  follow- 
ing more  general  result  : 

*  VI.  If  f(x)  denotes  any  polynomial  in  x  with  coefficients 
which  are  integers  (positive  or  zero  or  negative)  and  if  further 
a=b  mod  m,  then 

f(a)  =f(b)  mod  m. 

§  19.    Solutions  of  Congruences  by  Trial 

Let  f(x)  be  any  polynomial  in  x  with  coefficients  which 
are  integers  (positive  or  negative  or  zero).  Then  if  x  and  c 
are  any  two  integers  it  follows  from  the  last  theorem  of  the 
preceding  section  that 

f(x)  =f(x-\-cm)  mod  m.  (i) 

Hence  if  a  is  any  value  of  x  for  which  the  congruence 

/(#)=omodra  (2) 

is  satisfied,  then  the  congruence  is  also  satisfied  for  x=a-\-cm, 
where  c-is  any  integer  whatever.  The  numbers*  a -\-cni  are 
said  to  form  a  solution  (or  to  be  a  root)  of  the  congruence,  c 
being  a  variable  integer.  Any  one  of  the  integers  a-\-cm  may 
be  taken  as  the  representative  of  the  solution.  We  shall  often 
speak  of  one  of  these  numbers  as  the  solution  itself. 

Among  the  integers  in  a  solution  of  the  congruence  (2) 
there  is  evidently  one  which  is  positive  and  not  greater  than 
m.  Hence  all  solutions  of  a  congruence  of  the  type  (2)  may 
be  found  by  trial,  a  substitution  of  each' of  the  numbers  1,  2, 
.  .  .  ,  m  being  made  for  x.  It  is  clear  also  that  m  is  the  maxi- 
mum number  of  solutions  which  (2)  can  have  whatever  be 
the  function  f(x) .  By  means  of  an  example  it  is  easy  to  show 
that  this  maximum  number  of  solutions  is  not  always  possessed 
by  a  congruence*;  in  fact,  it  is  not  even  necessary  that  the 
congruence  bave  a  solution  at  all. 

This  is  illustrated  by  the  example 

x2  —  3=0  mod  5. 


40  THEORY   OF   NUMBERS 

In  order  to  show  that  no  solution  is  possible  it  is  necessary  to 
make  trial  only  of  the  values  i,  2,  3,  4,  5  for  x.  A  direct  sub- 
stitution verifies  the  conclusion  that  none  of  them  satisfies 
the  congruence;  and  hence  that  the  congruence  has  no  solution 
at  all. 

On  the  other  hand  the  congruence 

x5  —  x=o  mod  5 

has  the  solutions  x=i,  2,  3,  4,  5  as  one  readily  verifies;  that 
is,  this  congruence  has  five  solutions — the  maximum  number 
possible  in  accordance  with  the  results  obtained  above. 

EXERCISES 

1.  Show  that 

(a+b)v  =  av+bp.modp 

where  a  and  b  are  any  integers  and  p  is  any  prime. 

2.  From  the  preceding  result  prove  that 

a?  =a  mod  p 
for  every  integer  a. 

3.  Find    all    the    solutions    of    each    of    the    congruences    xn=x  mod  11, 
x10=i  mod  n,  xb  =  i  mod  n.  * 


§  20.  Properties  of  Congruences  Relative  to  Division 

The  properties  of  congruences  relative  to .  addition,  sub- 
traction and  multiplication  are  entirely  analogous  to  the  prop- 
erties of  algebraic  equations.  But  the  properties  relative  to 
division  are  essentially  different.     These  we  shall  now  give. 

I.  If  two  numbers  are  congruent  modulo  m  they  are  con- 
gruent modulo  d,  where  d  is  any  divisor  of  m. 

For,  from  o=5  mod  m,  we  have  a  =  b-\-cm  —  b-\-cfd.  Hence 
a=b  mod  d. 

II.  If  two  numbers  are  congruent  for  different  moduli  they 
are  congruent  for  a  modulus  which  is  the  least  common  multiple 
of  the  given  moduli. 


ELEMENTARY   PROPERTIES   OF   CONGRUENCES  41 

The  proof  is  obvious,  since  the  difference  of  the  given  num- 
bers is  divisible  by  each  of  the  moduli. 

III.  When  the  two  members  of  a  congruence  are  multiples  of 
an  integer  c  prime  to  the  modulus,  each  member  of  the  congruence 
may  be  divided  by  c. 

For,  if  ca=cb  mod  m  then  ca  —  cb  is  divisible  by  m.  Since 
c  is  prime  to  m  it  follows  that  a  —  b  is  divisible  by  m.  Hence 
a =b  mod  m. 

IV.  //  the  two  members  of  a  congruence  are  divisible  by  an 
integer  c,  having  with  the  modulus  the  greatest  common  divisor  5, 
one  obtains  a  congruence  equivalent  to  the  given  congruence  by 
dividing  the  two  members  by  c  and  the  modulus  by  5. 

By  hypothesis 

ac=bc  mod  m,     c=8ci,     m—hm\. 

Hence  c(a  —  b)  is  divisible  by  m.  A  necessary  and  sufficient 
condition  for  this  is  evidently  that  c\{a  —  b)  is  divisible  by  mi. 
This  leads  at  once  to  the  desired  result. 


§21.  Congruences  with  a  Prime  Modulus 

The  congruence  * 

aoxn+aixn~l-\-  .  .  .  +an=omod  p,     ao^omod  p, 

where  p  is  a  prime  number  and  the  a's  are  any  integers,  has  not 
more  than  n  solutions. 

Denote  the  first  member  of  this  congruence  by  f(x)  so  that 
the  congruence  may  be  written 

f(x)  =o  mod  p.  (i) 

Suppose  that  a  is  a  root  of  the  congruence,  so  that 

f(a)  =o  mod  p. 
Then  we  have 

f(x)  ^f(x)-f(a)  mod  p. 


* 


*  The  sign  ^  is  read  is  not  congruent  to. 


42  THEORY   OF   NUMBERS 

But,  from  algebra.  f(x)—f(a)  is  divisible  by  x  —  a.  Let  (x  —  a)a 
be  the  highest  power  of  x  —  a  contained  in  f(x)  —f(a) .  Then 
we  may  write 

f(x)-f(a)  =  (x-a)«f1(x),  (2) 

where /i(x)  is  evidently  a  polynomial  with  integral  coefficients. 
Hence  we  have 

}{x)  =  (x-a) a/i  0)  mod  p.  (3) 

We  shall  say  that  a  occurs  a  times  as  a  solution  of  (1);  or  that 
the  congruence  has  a  solutions  each  equal  to  a. 

Now  suppose  that  congruence  (1)  has  a  root  b  such  that 
b=£a  mod  p.     Then  from  (3)  we  have 

f(b)m(b^a)afi(b)modp. 

But  f(b)=omodp,     (b-a)a^omodp. 

Hence,  since  p  is  a  prime  number,  we  must  have 

fi(b)  =0  mod  p. 

By  an  argument  similar  to  that  just  used  above,  we  may 
show  that  /1  (x)  — /1  (b)  may  be  written  in  the  form 

where  /5  is  some  positive  integer.     Then  we  have 

f(x)  =  (x-a)a(x-byf2{x)  mod  p. 

Now  this  process  can  be  continued  until  either  all  the 
solutions  of  (1)  are  exhausted  or  the  second  member  is  a  prod- 
uct of  linear  factors  multiplied  by  the  integer  a0.  In  the  for- 
mer case  there  will  be  fewer  than  n  solutions  of  (1),  so  that 
our  theorem  is  true  for  this  case.     In  the  other  case  we  have 

f(x)^a0(x-a)a(x-by  .  .  .  (>-/)xmod/>. 

We  have  now  n  solutions  of  (1) :  a  counted  a  times,  b  counted 
0  times,  .  .  .  ,  I  counted  X  times;  «+/?+  .  .  .   +X  =  w. 


ELEMENTARY   PROPERTIES    OF   CONGRUENCES  43 

Now  let  77  be  any  solution  of  (i).     Then 

f(rl)=ao(ri-a)a(r,-b)f'  .  .  .  (v-l)x=omodp. 

Since  p  is  prime  it  follows  now  that  some  one  of  the  factors 
7}  —  a,  r]  —  b,  .  .  .  ,  t)—1  is  divisible  by  p.     Hence  77  coincides 
with  one  of  the  solutions  a,  b,  c,  .  .  .  ,  I.     That  is,   (1)  can 
have  only  the  n  solutions  already  found. 
This  completes  the  proof  of  the  theorem. 

EXERCISES 

1.  Construct  a  congruence  of  the  form 

aoxn-\-alxn~1-\-.  .  .  -\-an=omodm,     a0^omodf», 

having  more  than  n  solutions  and  thus  show  that  the  limitation  to  a  prime  mod- 
ulus in  the  theorem  of  this  section  is  essential. 

2.  Prove  that 

x*-i=(x-i)(x-2)(x-3)(x-4)(x-s)(x-6)  mod  7 
for  every  integer  x. 

3.  How  many  solutions  has  the  congruence  x5=i  mod  11?    the  congruence 
£5=2  mod  n? 


§  22.  Linear  Congruences 

From  the  theorem  of  the  preceding  section  it  follows  that 
the  congruence 

ax  =c  mod  p,     a^o  mod  p, 

where  p  is  a  prime  number,  has  not  more  than  one  solution. 
In  this  section  we  shall  prove  that  it  always  has  a- solution. 
More  generally,  we  shall  consider  the  congruence 

ax=c  mod  m 

where  m  is  any  integer.  The  discussion  will  be  broken  up 
into  parts  for  convenience  in  the  proofs. 

I.  The  congruence 

ax=\  mod  m,  (1) 


44  THEORY   OF   NUMBERS 

in  which  a  and  m  are  relatively  prime,  has  one  and  only  one  solu- 
tion. 

The  question  as  to  the  existence  and  number  of  the  solu- 
tions of  (i)  is  equivalent  to  the  question  as  to  the  existence 
and   number   of   integer   pairs   x,   y   satisfying   the   equation, 

ax— my  =  i,  (2) 

the  integers  x  being  incongruent  modulo  m.  Since  a  and  m 
are  relatively  prime  it  follows  from  theorem  IV  of  §  9  that 
there  exists  a  solution  of  equation  (2).  Let  x=a  and  y  =  p 
be  a  particular  solution  of  (2)  and  let  x  =  a  and  y  =  ]3  be  any 
solution  of  (2).     Then  we  have 

aa  —  mP  =  i, 
aa—mfi=i; 
whence 

a(a  —  a)—m(p  —  ~(3)  =0. 

Hence  a  — a  is  divisible  by  m,  since  a  and  m  are  relatively  prime. 
That  is,  a  =  amodm.  Hence  a  and  a  are  representatives  of 
the  same  solution  of  (1).  Hence  (1)  has  one  and  only  one 
solution,  as  was  to  be  proved. 

II.  The  solution  x=a  of  the  congruence  w=imodw,  in 
which  a  and  m  are  relatively  prime,  is  prime  to  m. 

For,  if  aa  — 1  is  divisible  by  m,  a  is  divisible  by  no  factor 
of  m  except  1. 

III.  The  congruence 

flx=cmodm  (3) 

in  which  a  and  m  and  also  c  and  m  are  relatively  prime,  has  one 
and  only  one  solution. 

Let  x  =  y  be  the  unique  solution  of  the  congruence 
a=i  mod  w.  Then  we  have  ayx=cy  =  imodm.  Now*  by 
I  we  see  that  there  is  one  and  only  one  solution  of  the  con- 
gruence ayx  =  imo<im\  and  from  this  the  theorem  follows  at 
once. 

Suppose  now  that  a  is  prime  to  m  but  that  c  and  m  have 
the  greatest  common  divisor  8  which  is  different  from  1 .  Then 
it  is  easy  to  see  that  any  solution  x  of  the  congruence  ax=c  mod  m 


ELEMENTARY   PROPERTIES   OF   CONGRUENCES  45 

must  be  divisible  by  <5.  The  question  of  the  existence  of  solu- 
tions of  the  congruence  ax=c  mod  m  is  then  equivalent  to  the 
question  of  the  existence  of  solutions  of  the  congruence 

x     c         t  m 

a  —  sb  -  mod  — , 

8      8  8 

where  x/8  is  the  unknown  integer.  From  III  it  follows  that 
this  congruence  has  a  unique  solution  x/8=a.  Hence  the 
congruence  ax^cmodm  has^the  unique  solution  x=8a.  Thus 
we  have  the  following  theorem: 

IV.  The  congruence  ax=cmodm,  in  which  a  and  m  are 
relatively  prime,  has  one  and  only  one  solution. 

Corollary.  The  congruence  ax=c  mod  p,  a=|=o  mod  p, 
where  p  is  a  prime  number,  has  one  and  only  one  solution. 

It  remains  to  examine  the  case  of  the  congruence  ax=(?mod  m 
in  which  a  and  m  have  the  greatest  common  divisor  d.  It  is 
evident  that  there  is  no  solution  unless  c  also  contains  this 
divisor  d.l  Then  let  us  suppose  that  a  =  ad,  c  =  yd,  m  =  txd. 
Then  for  every  x  such  that  ax=c  mod  m  we  have  ax =7  mod  ft; 
and  conversely  every  x  satisfying  the  latter  congruence  also 
satisfies  the  former.  Now  ax=yraodn  has  only  one  solu- 
tion. Let  j3  be  a  non-negative  number  less  than  jjl  which  satis- 
fies the  congruence  ax =7  mod  /*.  All  integers  which  satisfy 
this  congruence  are  then  of  the  form  fi+ixv,  where  v  is  an  integer. 
Hence  all  integers  satisfying  the  congruence  ax=cmodm  are 
of  the  form  P+jjiv;  and  every  such  integer  is  a  representative 
of  a  solution  of  this  congruence.     It  is  clear  that  the  numbers 

0,      0+M,      P  +  2H,    .    .    .    ,      fi  +  (d-l)ii  (A) 

are  incongruent  modulo  m  while  every  integer  of  the  form 
fi+nvis  congruent  modulo  m  to  a  number  of]the  set  (-4).  Hence 
the  congruence  ax=c  mod  m  has  the  d  solutions  (^4). 

This  leads  us  to  an  important  theorem  which  includes  all 
the  other  theorems  of  this  section  as  special  cases.  It  may  be 
stated  as  follows: 

V.  Let 

flx=cmodw 


46  THEORY  OF   NUMBERS 

be  any  linear  congruence  and  let  a  and  m  have  the  greatest  common 
divisor  d  (d^-i).  Then  a  necessary  and  sufficient  condition  for 
the  existence  of  solutions  of  the  congruence  is  that  c  be  divisible 
by  d.  If  this  condition  is  satisfied  the  congruence  has  just  d  solu- 
tions^ and  all  the  solutions  are  congruent  modulo  m/d. 

EXERCISES 

J  $ 

i.  Find  the  remainder  when  240  is  divided  by  31;   when  2 43  is  divided  by  31. 

2.  Show  that  228+i  has  the  factor  641. 

3.  Prove  that  a  number  is  a  multiple  of  9  if  and  only  if  the  sum  of  its  digits 
is  a  multiple  of  9. 

4.  Prove  that  a  number  is  a  multiple  of  n  if  and  only  if  the  sum  of  the  digits 
in  the  odd  numbered  places  diminished  by  the  sum  of  the  digits  in  the  even 
numbered  places  is  a  multiple  of  11. 


CHAPTER  IV 

THE  THEOREMS  OF  FERMAT  AND  WILSON 

§  23.  Fermat's  General  Theorem 
Let  m  be  any  positive  integer  and  let 

ai,     fl2,  •  •  •  ,  a*(m)  (A) 

be  the  set  of  <f>(m)  positive  integers  not  greater  than  m  and 
prime  to  m.  Let  a  be  any  integer  prime  to  m  and  form  the  set 
of  integers 

aai,    aa2,  .  .  .  ,     aa^m).  (B) 

No  number  aat  of  the  set  (B)  is  congruent  to  a  number  aa$ 
unlessy  =  z;  for,  from 

aai^=  aaj  mod  m 

we  have  Oi= a j  mod.  m\  whence  a%  =  ah  since  both  a\  and  aj 
are  positive  and  not  greater  than  m.  Theref ore  j  —  i .  Further- 
more, every  number  of  the  set  (B)  is  congruent  to  some  number 
of  the  set  (A).    Hence  we  have  congruences  of  the  form 

aai^a^  mod  m, 
aa2  =  au  mod  my 


00*<m)=a«0(w)mod*» 


No j^oriu|jfrers  in  the  second  members  are  equal,  since  aa^aaj 
un[^^t=j.  Hence  the  numbers  atl,  aiv  .  .  ,  ,  at  (m)  are 
the  numbers  #i,  #2,  ...  ,  a0(m)  in  some  order.  Therefore, 
if  we  multiply  the  above  system  of  congruences  together  mem- 

47 


48  THEORY   OF   NUMBERS 

ber  by  member  and  divide  each  member  of  the  resulting  con- 
gruence by  ai-d2  .  .  .  a^m)  (which  is  prime  to  m),  we  have 

This  result  is  known  as  Fermat's  general  theorem.     It  may 
be  stated  as  follows : 

If  m  is  any  positive  integer  and  a  is  any  integer  prime  to  m, 
then 

Corollary  i.     If  a  is  any  integer  not  divisible  by  a  prime 
number  p,  then 

ap~1  =  i  mod  p. 

Corollary  2.      If  p  is  any  prime  number   and  a  is  any 
integer,  then 

ap=a  mod  p. 


§  24.  Euler's  Proof  of  the  Simple  Fermat  Theorem\ 

The  theorem  of  Cor.  1,  §  23,  is  often  spoken  of  as  the  simple 
Fermat  theorem.  It  was  first  announced  by  Fermat  in  1679, 
but  without  proof.  The  first  proof  of  it  was  given  by  Euler 
in  1736.     This  proof  may  be  stated  as  follows: 

From  the  Binomial  Theorem  it  follows  readily  that 

(a+i)p=ap+imodp 
since 

p\ 


r\(p-r)\ 


,     o<r<p, 


is  obviously  divisible  by  p.     Subtracting  a-\-i  %>m  each  side 
of  the  foregoing  congruence,  we  have 

(a  +  i)p-(>  +  i)=ap-arnod£. 


THE   THEOREMS    OF   FERMAT   AND   WILSON  49 

Hence  if  av  —  a  is  divisible  by  p,  so  is  (a  +  i)p  —  (a+i).  But 
ip  —  i  is  divisible  by  p.  Hence  2P  —  2  is  divisible  by  p\  and 
then  3P  — 3;  and  so  on.     Therefore,  in  general,  we  have 

ap=a  mod  p. 

If  a  is  prime  to  p  this  gives  ap~x  =  1  mod  ^,  as  was  to  be  proved. 

If  instead  of  the  Binomial  Theorem  one  employs  the  Poly- 
nomial Theorem,  an  even  simpler  proof  is  obtained.  For, 
from  the  latter  theorem,  we  have  readily 

'(ai+a2+   •   .   .    +aa)p=aip+a2p+   .   .   .    +aap  mod  p. 

Putting  ai=a2=  .  .  .   =aa=i     we  have 

ap=a  mod  p, 

from  which  the  theorem  follows  as  before. 


§  25.  Wilson's  Theorem 

From  the  simple  Fermat  theorem  it  follows  that  the  con- 
gruence 

xp~1  =  i  mod^ 

has  the  p  —  i  solutions  1,  2,  3,  .  .  .  ,  p  —  i.    Hence  from  the 
discussion  in  §  21  it  follows  that 


xp     —  i==(x  —  i)(x  —  2)  .  .  .  (x  —  p  —  i)  mod  p, 

this  relation  being  satisfied  for  every  value  of  x.    Putting  x=o 
we  have 

(_I)  =  (_I)p-i.Iv2.3  ,  ,  .  p-imodp. 

If  p  is  an  odd  prime  this  leads  to  the  congruence 


i- 2-3  .  .  .  p  —  i-\-i=omod  p. 

Now  for  p  =  2  this  congruence  is  evidently  satisfied.    Hence 
we  have  the  Wilson  theorem : 

Every  prime  number  p  satisfies  the  relation 

i* 2*3  .  .  .  p  —  i  +  i=omod^. 


50  THEORY    OF    NUMBERS 

An  interesting  proof  of  this  theorem  on  wholly  different 
principles  may  be  given.  Let  p  points  be  distributed  at  equal 
intervals  on  the  circumference  of  a  circle.  The  whole  number 
of  ^-gons  which  can  be  formed  by  joining  up  these  p  points 
in  every  possible  order  is  evidently 


i 


„      2p 


p(p-i)(p-2)  .  .  .  3-2-i; 


for  the  first  vertex  can  be  chosen  in  p  ways,  the  second  in  p  —  i 
ways,  .  .  .  ,  the  (p  —  i)th  in  two  ways,  and  the  last  in  one 
way;  and  in  counting  up  thus  we  have  evidently  counted  each 
polygon  2p  times,  once  for  each  vertex  and  for  each  direction 
from  the  verfcqx  around  the  polygon.  Of  the  total  number 
of  polygons  \{p  —  i)  are  regular  (convex  or  stellated)  so  that 
a  revolution  through  360°/^  brings  each  of  these  into  coin- 
cidence with  Its  former  position.  The  number  of  remaining 
p-gons  must  be  divisible  by  p\  for  with  each  such  ^>-gon  we  may 
associate  the  p  —  i  ^-gons  which  can  be  obtained  from  it  by 
rotating  it  through  successive  angles  of  360°/^.     That  is, 

-^-p(p-i)(p-2)  .  .  .  3-2-i--0-i)=omod/>. 

2p  2 

Hence 

(p  —  i)(p  —  2)  .  .  .  3-2-1  —  />+i=o  mod  p; 

and  from  this  it  follows  that 


1-2  ..  .  ^—1  +  1=0  mod  p, 
as  was  to  be  proved. 

§  26.  The  Converse  of  Wilson's  Theorem 

Wilson's  theorem  is  noteworthy  in  that  its  converse  is  also 
true.     The  converse  may  be  stated  as  follows: 
Every  integer  n  such  that  the  congruence 


1-2-3  .  .  .  n—  1  +  1=0  mod  n 
is  satisfied  is  a  prime  number. 


THE   THEOREMS   OF   FERMAT   AND   WILSON  51 

For,  if  n  is  not  prime,  there  is  some  divisor  d  of  n  different 
from  i  and  less  than  n.  For  such  a  d  we  have  1-2-3  •  •  • 
n— 1=0  mod  d;  so  that  1-2  ..  .  w-i  +  i^omodi;  and 
hence  1-2  .  .  .  n— 1  +  1=^0  mod  n.  Since  this, contradicts  our 
hypothesis  the  truth  of  the  theorem  follows. 

Wilson's  theorem  and  its  converse  may  be  combined  into 
the  following  elegant  theorem: 

A  necessary  and  sufficient  condition  that  an  integer  n  is  prime 

is  that  

1-2-3  •  •  •  w-i  +  i=omodw. 

Theoretically  this  furnishes  a  complete  and  elegant  test 
as  to  whether  a  given  number  is  prime.  But,  practically, 
the  labor  of  applying  it  is  so  great  that  it  is  useless  for  verifying 
large  primes. 


§27.  Impossibility  of  1-2-3  •  •  •  n~ i  +  i=w*  for  n>$. 

In  this  section  we  shall  prove  the  following  theorem: 
There  exists  no  integer  k  for  which  the  equation 


i-  2 


3  .  .  .  n—  i-\-i=n* 


is  true  when  n  is  greater  than  5. 

If  n  contains  a  divisor  d  different  from  1  and  n,  the  equa- 
tion is  obviously  false;  for  the  second  member  is  divisible 
by  d  while  the  first  is  not.  Hence  we  need  to  prove  the  theorem 
only  for  primes  n. 

Transposing  1  to  the  second  member  and  dividing  by  n  —  1 
we  have 

1-2-3  •  •  •  n  —  2=nt~1+nt~2+  .  .  .  +n+i. 

If  w>5  the  product  on  the  left  contains  both  the  factor  2  and 
the  factor  \{n  —  1);  that  is,  the  first  member  contains  the  fac- 
tor n—  1.  But  the  second  member  does  not  contain  this  fac- 
tor, since  for  n  =  i  the  expression  w*_1+  .  .  .  -\-n-\-i  is  equal 
to  kj^o.     Hence  the  theorem  follows  at  once. 


52  theory  of  numbers 

§  28.  Extension  of  Fermat's  Theorem 

The  object  of  this  section  is  to  extend  Fermat's  general 
theorem  and  incidentally  to  give  a  new  proof  of  it.  We  shall 
base  this  proof  on  the  simple  Fermat  theorem,  of  which  we 
have  already  given  a  simple  independent  proof.  This  theorem 
asserts  that  for  every  prime  p  and  integer  a  not  divisible  by  p, 
we  have  the  congruence 

ap~1  =  i  mod  p. 

Then  let  us  write 

a*-1  =  i+hp.  (1) 

Raising  each  member  of  this  equation   to  the  pth  power  we 
may  write  the  result  in  the  form 

aviV-D    =I+hlp2  (2) 

where  hi  is  an  integer.     Hence 

^-"simod^, 

By  raising  each  member  of  (2)  to  the  p    power  we  can  readily 
show  that 

/^^imod^3. 

It  is  now  easy  to  see  that  we  shall  have  in  general 

wUere  a  is  a  positive  integer;  that  is, 

a*(pa)  =  1  m0(i  p«m 

For  the  special  case  when  p  is  2  this  result  can  be  extended. 
For  this  case  (1)  becomes 

a  =  1  +  2/2. 
Squaring  we  have 

<z2  =  i+4/*0+i). 
Hence, 

a2  =  i+8Ai,  (3) 


THE   THEOREMS   OF   FERMAT   AND   WILSON  53 

where  h\  is  an  integer.     Therefore 

a2  =  i  mod  23. 
Squaring  (3)  we  have 

a22  =  i  +  2*h2; 
a22  =  1  mod  24. 

It  is  now  easy  to  see  that  we  shall  have  in  general 
ifa>2.     That  is, 


a2"     =  1  mod  2a 


a^2a)  =  i  mod  2a  if  «>2. 

Now  in  terms  of  the  ^-function  let  us  define  a  new  function 
X(m)  as  follows: 

\(2a)  =  <£(2aX    if    a  =  o,  1,  2; 

X(2«)=^(2a)      if     a>2- 

\(pa)  =  <i>(pa)     if  p  is  an  odd  prime; 
\{2ap^p2^  .  .  .  pnan)       =M, 

where  M  is  the  least  common  multiple  of 

x(2«),    x(/>!«0,    x(/>2««),  .  .  .  ,    X(/>.««), 

2,  ^1,  p2,  -  .  .  ,  pn  being  different  primes. 
Denote  by  m  the  number 

rn  =  2ap1aip2a*  .  .  .  pnan. 

Let  a  be  any  number  prime  to  m.     From  our  preceding  results 
we  have 

#(*■>«!  mod '?«." 

bWsi  mod#i% 
ax(p2«2)  =  x  moc]  ^2«s 


aX(P„«»)  =  lmod^an> 


54  THEORY   OF   NUMBERS 

Now  any  one  of  these  congruences  remains  true  if  both  of 
its  members  are  raised  to  the  same  positive  integral  power, 
whatever  that  power  may  be.  Then  let  us  raise  both  members 
of  the  first  congruence  to  the  power  \(m)/\(2a);  both  members 
of  the  second  congruence  to  the  power  X(w) /Mpi"1)  5  •  •  •  J 
both  members  of  the  last  congruence  to  the  power  \(m)/\(pnan). 
Then  we  have 

a\(m)  _  x  mo(J  2a} 


From  these  congruences  we  have  immediately 

a\(m)  m  j  mocJ  Mf 

We  may  state  this  result  in  full  in  the  following  theorem : 

//  a  and  m  are  any  two  relatively  prime  positive  integers,  the 

congruence 

aHm)  s  j  m0(j  m 

is  satisfied. 

As  an  excellent  example  to  show  the  possible  difference 
between  the  exponent  \(m)  in  this  theorem  and  the  exponent 
4>(m)  in  Fermat's  general  theorem,  let  us  take 

w  =  26-33-5-7-i3-i7-i9'37-73. 
Here 

X(»=24-32,     4>0)  =  221-310. 

In  a  later  chapter  we  shall  show  that  there  is  no  exponent 
v  less  than  \(m)  for  which  the  congruence 

a"  =  i  mod  m 

is  verified  for  every  integer  a  prime  to  m. 

From  our  theorem,  as  stated  above,  Fermat's  general  the- 
orem follows  as  a  corollary,  since  X(m)  is  obviously  a  factor 
of  4>(m), 

<Km)  =  <K2«)*(/>i«0  •  ■  •  4>(£»an). 


THE   THEOREMS   OF   FERMAT   AND   WILSON  55 

EXERCISES 

i.  Show  that  a16=i  mod  16320,  for  every  a  which  is  prime  to  16320. 
2.  Show  that  a12  =  i  mod  65520,  for  every  a  which  is  prime  to  65520. 
3*.  Find  one  or  more  composite  numbers  P  such  that 

ap~l  =  i  modP 

for  every  a  prime  to  P.     (Compare  this  problem  with  the  next  section.) 

§  29.  On  the  Converse  of  Fermat's  Simple  Theorem 

The  fact  that  the  converse  of  Wilson's  theorem  is  a  true 
proposition  leads  one  naturally  to  inquire  whether  the  con- 
verse of  Fermat's  simple  theorem  is  true.  Thus,  we  may  ask  the 
question:  Does  the  existence  of  the  congruence  2n_1  =  i  mod  n 
require  that  n  be  a  prime  number?  The  Chinese  answered 
this  question  in  the  affirmative  and  the  answer  passed  unchal- 
lenged among  them  for  many  years.  An  example  is  sufficient 
to  show  that  the  theorem  is  not  true.     We  shall  show  that 

234o  =  j  mocj  241 

although  341,  =  11 -31,  is  not  a  prime  number.  Now  210— i 
=  3-11-31.  Hence  210=imod34i.  Hence  2340=i  mod  341. 
From  this  it  follows  that  the  direct  converse  of  Fermat's  the- 
orem is  not  true.  The  following  theorem,  however,  which  is 
a  converse  with  an  extended  hypothesis,  is  readily  proved. 

If  there  exists  an  integer  a  such  that 
an~l  =  i  mod  n 

and  if  further  there  does  not  exist  an  integer  v  less  than  n—i  such 

that 

av=i  mod  n, 

then  the  integer  n  is  a  prime  number. 

For,  if  n  is  not  prime,  <f>(n)<n—i.     Then  for  v  =  <f>(n)  we 
have  a"  =  1  mod  n,  contrary  to  the  hypothesis  of  the  theorem. 


56  THEORY   OF   NUMBERS 


§30.  Application  of   Previous  Results  to  Linear   Con- 
gruences 

The  theorems  of  the  present  chapter  afford  us  a  ready  means 
of  writing  down  a  solution  of  the  congruence 

ax=c  mod  m. 

We  shall  consider  only  the  case  in  which  a  and  m  are  relatively 
prime,  since  the  general  case  is  easily  reducible  to  this  one, 
as  we  saw  in  the  preceding  chapter. 

Since  a  and  m  are  relatively  prime  we  have  the  congruences 
Hence  either  of  the  numbers  x, 


is  a  representative  of  the  solution  of  (1).     Hence  the  following 
theorem : 

If  ax=c  mod  m 

is  any  linear  congruence  in  which  a  and  m  are  relatively  prime, 
then  either  of  the  numbers  x, 

x  =  cax{m)-1,     x  =  ca^m)-\ 

is  a  representative  of  the  solution  of  the  congruence. 

The  former  representative  of  the  solution  is  the  more  con- 
venient of  the  two,  since  the  power  of  a  is  in  general  much  less 
in  this  case  than  in  the  other. 

EXERCISE 

Find  a  solution  of  ^x  =  ^  mod  26-3«5-i7.     Note  the  greater  facility  in  apply- 
ing the  first  of  the  above  representatives  of  the  solution  rather  than  the  second. 


the  theorems  of  fermat  and  wilson  57 

§  31.  Application  of  the  Preceding  Results  to  the  Theory 
of  Quadratic  Residues 

In  this  section  we  shall  apply  the  preceding  results  of  this 
chapter  to  the  problem  of  rinding  the  solutions  of  congruences 
of  the  form 

az2+/3z+Y=o  mod  /z 

where  a,  /3,  7,  fi  are  integers.     These  are  called  quadratic  con- 
gruences. 

The  problem  of  the  solution  of  the  quadratic  congruence 
(1)  can  be  reduced  to  that  of  the  solution  of  a  simpler  form  of 
congruence  as  follows:  Congruence  (1)  is  evidently  equivalent 
to  the  congruence 

4«222  -f  4^2+40:7  =  0  mod  4.afi.  (1) 

But  this  may  be  written  in  the  form 

(2az+(3)2  =  (32— 40:7  mod  4<xfjL. 
Now  if  we  put 

2a2+/3=xmod  4a/x  (2) 

and 

|82 — 4«7  =  a,    4ctfjL  =  m, 
we  have 

x2  =  amodm.  (3) 

We  have  thus  reduced  the  problem  of  solving  the  general  con- 
gruence (1)  to  that  of  solving  the  binomial  congruence  (3) 
and  the  linear  congruence  (2).  The  solution  of  the  latter  may 
be  effected  by  means  of  the  results  of  §  30.  We  shall  there- 
fore confine  ourselves  now  to  a  study  of  congruence  (3).  We 
shall  make  a  further  limitation  by  assuming  that  a  and  m 
are  relatively  prime,  since  it  is  obvious  that  the  more  general 
case  is  readily  reducible  to  this  one. 
The  example 

#2  =  3  mod  5 


58  THEORY   OF    NUMBERS 

shows  at  once  that  the  congruence  (3)  does  not  always  have  a 
solution.  First  of  all,  then,  it  is  necessary  to  find  out  in  what 
cases  (3)  has  a  solution.  Before  taking  up  the  question  it  will 
be  convenient  to  introduce  some  definitions. 

Definitions.  An  integer  a  is  said  to  be  a  quadratic 
residue  modulo  m  or  a  quadratic  non-residue  modulo  m  accord- 
ing as  the  congruence 

#2  =  amodw 

has  or  has  not  a  solution.  We  shall  confine  our  attention  to 
the  case  when  m  >  2 . 

We  shall  now  prove  the  following  theorem : 

I.  If  a  and  m  are  relatively  prime  integers,  a  necessary  con- 
dition that  a  is  a  quadratic  residue  modulo  m  is  that 

ahMm)  _  j  mo(J  m 

Suppose  that  the  congruence  x2  =  a  mod  m  has  the  solu- 
tion x=a.     Then  a2  =  a  mod  m.     Hence 

aX<m)  m  aJX(m)  mod  m 

Since  a  is  prime  to  m  it  is  clear  from  a2  =  amod  m  that  a  is  prime 
to  m.     Hence  ax(w)  m  1  mod  m.     Therefore  we  have 

I=a§M«)  modra. 

That  is,  this  is  a  necessary  condition  in  order  that  a  shall  be 
a  quadratic  residue  modulo  m. 

In  a  similar  way  one  may  prove  the  following  theorem: 

II.  If  a  and  m  are  relatively  prime  integers,  a  necessary  con- 
dition that  a  is  a  quadratic  residue  modulo  m  is  that 

ai4im)  =  i  mod  m. 

When  m  is  a  prime  number  p  each  of  the  above  results 
takes  the  following  form:  If  a  is  prime  to  p  and  is  a  quadratic 
residue  modulo  p,  then 


THE    THEOREMS    OF    FERMAT    AND    WILSON  59 

We  shall  now  prove  the  following  more  complete  theorem, 
without  the  use  of  I  or  II. 

III.  If  p  is  an  odd  prime  number  and  a  is  an  integer  not 
divisible  by  p,  then  a  is  a  quadratic  residue  or  a  quadratic  non- 
residue  modulo  p  according  as 

aKp-D  =  +I     or     ^-i)=_imod^ 

This  is  called  Euler's  criterion. 

Given  a  number  a,  not  divisible  by  p,  we  have  to  determine 
whether  or  not  the  congruence 

x2  =  a  mod  p 

has  a  solution.     Let  r  be  any  number  of  the  set 

i,  2,  3,  ...  ,  p-i  (A) 

and  consider  the  congruence 

rx  =  amod  p. 

This  has  always  one  and  just  one  solution  x  equal  to  a  number 
5  of  the  set  (.4).  Two  cases  can  arise  :T  either  for  every  r  of  the 
set  (A)  the  corresponding  s  is  different  from  r  orjfor  some  r 
of  the  set  (.4)  the  corresponding  5  is  equal  to  r.  The  former 
is  the  case  when  a  is  a-  quadratic  non-residue  modulo  p\  the 
latter  is  the  case  when  a  is  a  quadratic  residue  modulo  p.  We 
consider  the  two  cases  separately. 

In  the  first  case  the  numbers  of  the  set  (.4)  go  in  pairs  such 
that  the  product  of  the  numbers  in  the  pair  is  congruent  to  a 
modulo  p.  Hence,  taking  the  product  of  all  \{p  —  i)  pairs, 
we  have 

1-2-3  .  .  .  p  —  i  =  -]-a^p~1)  mod  p. 
But 

1-2-3  .  .  .  p  —  i  =  —  i  mod  p. 

Hence 

al(p-u  =  __j  mod  p^ 

whence  the  truth  of  one  part  of  the  theorem. 


60  THEORY   OF   NUMBERS 

In  the  other  case,  namely  that  in  which  some  r  and  corre- 
sponding s  are  equal,  we  have  for  this  r 

r2  =  a  mod  p 
and 

(p—r)2  =  a  mod  p. 

Since  x2=amod  p  has  at  most  two  solutions  it  follows  that 
all  the  integers  in  the  set  (A)  except  r  and  p—r  fall  in  pairs 
such  that  the  product  of  the  numbers  in  each  pair  is  congruent 
to  a  modulo  p.  Hence,  taking  the  product  of  all  these  pairs, 
which  are  ^(p—i)  —  i  in  number,  and  multiplying  by  r{p—r) 
we  have 

1-2-3  .  .  .  p  —  i  =  (p— r)ra^p~l^~l  mod  p 
=  -r2ai(p-i)-imodp 

=  -a-ah{lp-l)-lmodp 
=  _ai(p-Dmo(i^. 


Since  1-2-3  •  •  •  P "~ 1  m r~ 1  m°d  P  we  have 

a^p-v  =  +imodp; 

whence  the  truth  of  another  part  of  the  theorem. 
Thus  the  proof  of  the  entire  theorem  is  complete. 


CHAPTER  V. 

PRIMITIVE  ROOTS  MODULO  m- 

§  32.  Exponent  of  an  Integer  Modulo  m 
Let 

01,  02,    ...    ,  0£(m)  (^) 

be  the  set  of  <j>(m)  positive  integers  not  greater  than  m  and 
prime  to  m;  and  let  a  denote  any  integer  of  the  set  (A).  Now 
any  positive  integral  power  of  a  is  prime  to  m  and  hence  is 
congruent  modulo  w  to  a  number  of  the  set  (A).  Hence, 
among  all  the  powers  of  0  there  must  be  two,  say  an  and  av, 
n>v,  which  are  congruent  to  the  same  integer  of  the  set  (A). 
These  two  powers  are  then  congruent  to  each  other;    that  is, 

0n=a"  mod  m. 

Since  0"  is  prime  to  m  the  members  of  this  congruence  may  be 
divided  by  0".     Thus  we  have 

an~"=i  mod  m. 

That  is,  among  the  powers  of  a  there  is  one  at  least  which  is 
congruent  to  1  modulo  m. 

Now,  in  the  set  of  all  powers  of  a  which  are  congruent  to 
1  modulo  m  there  is  one  in  which  the  exponent  is  less  than  in 
any  other  of  the  set.  Let  the  exponent  of  this  power  be  d, 
so  that  ad  is  the  lowest  power  of  a  such  that 

ad=  1  mod  m. 

61 


62  THEORY   OF   NUMBERS 

We  shall  now  show  that  if  aa=  i  mod  m,  then  a  is  a  multiple 
of  d.    Let  us  write 

a  =  d8+p,     o^(3<d. 
Then 

aa=imodm,  (2) 

ad8=i  mod  m,  (3) 

the  last  congruence  being  obtained  by  raising  (1)  to  the  power 
5.     From  (3)  we  have 

ads+p=ap  mod  m; 
or 

ap=i  mod  m. 

Hence  /3  =  o,  for  otherwise  d  is  not  the  exponent  of  the  lowest 
power  of  a  which  is  congruent  to  1  modulo  m.  Hence  d  is  a 
divisor  of  a. 

These  results  may  be  stated  as  follows: 
I.  If  m  is  any  integer  and  a  is  any  integer  prime  to  m,  then 
there  exists  an  integer  d  such  that 

ad  =  1  mod  m 
while  there  is  no  integer  /3  less  than  d  for  which 

a&=i  mod  m. 
Further,  a  necessary  and  sufficient  condition  that 

a"=i  mod  m 
is  that  v  is  a  multiple  of  d. 

DEFINITION.  The  integer  d  which  is  thus  uniquely  deter- 
mined when  the  two  relatively  prime  integers  a  and  m  are  given 
is  called  the  exponent  of  a  modulo  m.  Also,  d  is  said  to  be 
the  exponent  to  which  a  belongs  modulo  m. 

Now,  in  every  case  we  have 


PRIMITIVE   ROOTS   MODULO   M  63 

if  a  and  m  are  relatively  prime.     Hence  from  the  preceding 
theorem  we  have  at  once  the  following: 

II.  The  exponent  d  to  which  a  belongs  modulo  m  is  a  divisor 
of  both  <f>(m)  and  \(m). 


§  33.  Another  Proof  of  Fermat's  General  Theorem 

In  this  section  we  shall  give  an  independent  proof  of  the 
theorem  that  the  exponent  d  of  a  modulo  m  is  a  divisor  of  <j>(m) ; 
from  this  result  we  have  obviously  a  new  proof  of  Fermat's 
theorem  itself. 

We  retain  the  notation  of  the  preceding  section.  We  shall 
first  prove  the  following  theorem : 

I.  The  numbers 

1,  a,  a2,  ...  ,  a*-1  (A) 

are  incongruent  each  to  each  modulo  m. 

For,  if  fl^^modw,  where  o<a<d  and  o<0<d,  a>ft, 
we  have  aa~^=i  mod  m,  so  that  d  is  not  the  exponent  to  which 
a  belongs  modulo  m,  contrary  to  hypothesis. 

Now  any  number  of  the  set  (A)  is  congruent  to  some  number 
of  the  set 

Hi,  42,  ...  ,  a*(TO).  (B) 

Let  us  undertake  to  separate  the  numbers   (B)  into  classes 

after  the  following  manner:    Let  the  first  class  consist  of  the 

numbers 

(I)  ao,  ai,  a2,   .   .   .  ,  «„_!, 

where  at  is  the  number  of  the  set  (B)  to  which  ai  is  congruent 
modulo  m. 

If  the  class  (I)  does  not  contain  all  the  numbers  of  the  set 
(B),  let  at  be  any  number  of  the  set  (B)  not  contained  in  (I) 
and  form  the  following  set  of  numbers: 

(IIr)  aoat,  aiat,  a2#«,   .   .   .  ,   ad_!a<. 


64  THEORY   OF   NUMBERS 

We  shall  now  show  that  no  number  of  this  set  is  congruent  to 
a  number  of  class  (I) .     For,  if  so,  we  should  have  a  congruence 

of  the  form 

ataj^aic  mod  m\ 
hence 

(Ha*  =  ak  mod  m, 
so  that 

aiad  =  ak+d~3  mod  m; 

or  <h  =  dk+d~j  mod  my 


so  that  at  would  belong  to  the  set  (I)  contrary  to  hypothesis. 

Now  the  numbers  of  the  set  (II')  are  all  congruent  to  num- 
bers of  the  set  (B) ;  and  no  two  are  congruent  to  the  same  num- 
ber of  this  set.  For,  if  so,  we  should  have  two  numbers  of 
(II')  congruent;  that  is,  akdi=ajdi  mod  m,  or  ak=aj  mod  m; 
and  this  we  have  seen  to  be  impossible. 

Now  let  the  numbers  of  the  set  (B)  to  which  the  numbers 
of  the  set  (II')  are  congruent  be  in  order  the  following: 

(II)  00,    01,    02,     ...    ,    0d-!- 

These  numbers  constitute  our  class  (II). 

If  classes  (I)  and  (II)  do  not  contain  all  the  numbers  of  the 
set  (B) ,  let  dj  be  a  number  of  the  set  (B)  not  contained  in  either 
of  the  classes  (I)  and  (II) :   and  form  the  set  of  numbers 

(III')  aodj,  aidj,  4x2a j,   .   .   .   ,  ad_1dJ. 

Just  as  in  the  preceding  case  it  may  be  shown  that  no  number 
of  this  set  is  congruent  to  a  number  of  class  (I)  and  that  the 
numbers  of  (III')  are  incongruent  each  to  each.  We  shall 
also  show  that  no  number  of  (III')  is  congruent  to  a  number 
of  class  (II).  For,  if  so,  we  should  have  dkdj=(3i  mod  m.  Hence 
akdj=dldi  mod  m\  or  dj=dl+d~kdi  mod  m,  from  which  it 
follows  that  dj  is  of  class  (II),  contrary  to  hypothesis. 

Now  let  the  numbers  of  the  set  (B)  to  which  the  numbers 
of  the  set  (III')  are  congruent  be  in  order  the  following: 

(HI)  TO,  71,  72,   •   •    •   ,   7d-i- 

These  numbers  form  our  class  (III). 


PRIMITIVE   ROOTS   MODULO   M  65 

It  is  now  evident  that  the  process  may  be  continued  until 
all  the  numbers  of  the  set  (B)  have  been  separated  into  classes, 
each  class  containing  d  integers,  thus: 

(i) 
(ii) 

(in) 

(      )  Xo,  Xi,  X2, 


OtO, 

ai, 

CL2,     ■ 

•    •    j    ad-l> 

Po, 

01, 

02,     . 

■    •    >    Pd-1> 

7o, 

71, 

72,     • 

•    •    ,    7d-l, 

^-1- 


The  set  (5),  which  consists  of  4>{m)  integers,  has  thus  been 
separated  into  classes,  each  class  containing  d  integers.  Hence 
we  conclude  that  d  is  a  divisor  of  <j>(m).  Thus  we  have  a  second 
proof  of  the  theorem: 

II.  If  a  and  m  are  any  two  relatively  prime  integers  and  d 
is  the  exponent  to  which  a  belongs  modulo  m,  then  d  is  a  divisor 
of  <t>(m). 

In  our  classification  of  the  numbers  (B)  into  the  rectangular 
array  above  we  have  proved  much  more  than  theorem  II; 
in  fact,  theorem  II  is  to  be  regarded  as  one  only  of  the  con- 
sequences of  the  more  general  result  contained  in  the  array. 

If  we  raise  each  member  of  the  congruence 

ad=i  mod  m 

to  the  (integral)  power  <f>(m)/d,  the  preceding  theorem  leads 
immediately  to  an  independent  proof  of  Fermat's  general 
theorem. 


§  34.  Definition  of  Primitive  Roots 

Definition.  Let  a  and  m  be  two  relatively  prime  integers. 
If  the  exponent  to  which  a  belongs  modulo  m  is  <f>(m),  a  is  said 
to  be  a  primitive  root  modulo  m  (or  a  primitive  root  of  m). 

In  a  previous  chapter  we  saw  that  the  congruence 

aHm)  =  j  mod  m 


66  THEORY   OF   NUMBERS 

is  verified  by  every  pair  of  relatively  prime  integers  a  and  m. 
Hence,  primitive  roots  can  exist  only  for  such  a  modulus  m  as 
satisfies  the  equation 

<t>(m)=\(m).  (i) 

We  shall  show  later  that  this  is  also  sufficient  for  the  existence 
of  primitive  roots. 

From  the  relation  which  exists  in  general  between  the 
^-function  and  the  X-function  in  virtue  of  the  definition  of  the 
latter,  it  follows  that  (i)  can  be  satisfied  only  when  m  is  a  prime 
power  or  is  twice  an  odd  prime  power. 

Suppose  first  that  m  is  a  power  of  2,  say  m  =  2a.  Then  (1)  is 
satisfied  only  if  a  =  o,  1,  2.  For  a  =  oor  1,  1  itself  is  a  primitive 
root.  For  a  =  2,  3  is  a  primitive  root.  We  have  therefore 
left  to  examine  only  the  cases 

m  =  pa,     m  =  2pa 

where  p  is  an  odd  prime  number.  The  detailed  study  of  these 
cases  follows  in  the  next  sections. 


§35.  Primitive  Roots  Modulo  p. 

We  have  seen  that  if  p  is  a  prime  number  and  d  is  the 
exponent  to  which  a  belongs  modulo  p,  then  J  is  a  divisor  of 
0(^)=^-i.     Now,  let 

di,  d2,  d3,  .  .  .  ,  dr 

be  all  the  divisors  of  p  —  1  and  let  $(dl)  denote  the  number  of 
integers  of  the  set 

1,  2,  3,  ...  ,  p-i 

which  belong  to  the  exponent  d%.     If  there  is  no  integer  of  the 
set  belonging  to  this  exponent,  then  \p(di)  =0. 


PRIMITIVE   ROOTS   MODULO   m  67 

Evidently  every  integer  of  the  set  belongs  to  some  one  and 
only  one  of  the  exponents  d\,  d2,  .  .  .  ,  dr.  Hence  we  have 
the  relation 

*(<*l)  +  *(*0  +    •    •    •     ++(dT)=p-I.  (l) 

But 

*(<Zl)+*(<fe)+    •    •    •    +<t>(dr)=p-I.  (2) 

If  then  we  can  show  that 

*(<*.)£*(<*.)  (3) 

for  i  =  i,  2,  .  .  .  ,  r,  it  will  follow  from  a  comparison  of  (i) 
and  (2)  that 

${di)  =  <t>{dt). 

Accordingly,  we  shall  examine  into  the  truth  of  (3). 
Now  the  congruence 

xd*  =  imodp  (4) 

has  not  more  than  di  roots.  If  no  root  of  this  congruence 
belongs  to  the  exponent  di,  then  \J/(di)=o  and  therefore  in  this 
case  we  have  \p(di)  <4>(di).  On  the  other  hand  if  a  is  a  root 
of  (4)  belonging  to  the  exponent  di,  then 

a,  a\  a*,  .  .  .  ,  ad*  (5) 

are  a  set  of  dt  incongruent  roots  of  (4) ;  and  hence  they  are  the 
complete  set  of  roots  of  (4). 

But  it  is  easy  to  see  that  a*  does  or  does  not  belong  to  the 
exponent  di  according  as  k  is  or  is  not  prime  to  di,  for,  if  a* 
belongs  to  the  exponent  /,  then  /  is  the  least  integer  such  that 
kt  is  a  multiple  of  di.  Consequently  the  number  of  roots  in 
the  set  (5)  belonging  to  the  exponent  di  is  <f>(di).  That  is, 
in  this  case  \j/(di)  =  4>(di) .  Hence  in  general  ^(dt)  £^(J«). 
Therefore  from  (1)  and  (2)  we  conclude  that 

t(di)  =  4>(di),  i=i,  2,  .  .  .  ,  r. 


68  THEORY   OF   NUMBERS 

The  result  thus  obtained  may  be  stated  in  the  form  of  the 
following  theorem: 

I.  If  p  is  a  prime  number  and  d  is  any  divisor  of  p  —  i ,  then 
the  number  of  integers  belonging  to  the  exponent  d  modulo  p 
is  <f)(d). 

In  particular: 

II.  There  exist  primitive  roots  modulo  p  and  their  number 
is  4>{p-i). 


§36.  Primitive  Roots  Modulo  pa,  p  an  Odd  Prime 

In  proving  that  there  exist  primitive  roots  modulo  pa,  where 
p  is  an  odd  prime  and  a>  1,  we  shall  need  the  following  theorem: 

I.  There  always  exists  a  primitive  root  y  modulo  p  for  which 
yp~l  —  1  is  not  divisible  by  p2. 

Let  g  be  any  primitive  root  modulo  p.  If  gp~1  —  1  is  not 
divisible  by  p2  our  theorem  is  verified.  Then  suppose  that 
gv~x  —  1  is  divisible  by  p2,  so  that  we  have 

where  k  is  an  integer.     Then  put 

7  =  g+Xp 
where  x  is  an  integer.     Then  7=g  mod  p,  and  hence 

yh  =  gh  mod  p; 

whence  we  conclude  that  7  is  a  primitive  root  modulo  p.     But 

I  !  2  I 

=  P(kp+tllr*x+(P-*)(p-*)f-Zx2p+    .    . 

Hence 

7p-1-i=^(-^"2x)  mod^2. 


PRIMITIVE   ROOTS    MODULO    M  t)9 

Therefore  it  is  evident  that  x  can  be  so  chosen  that  yp~1  —  i 
is  not  divisible  by  p2.  Hence  there  exists  a  primitive  root  7 
modulo  p  such  that  7P_1  —  1  is  not  divisible  by  p2.     Q.  E.  D. 

We  shall  now  prove  that  this  integer  7  is  a  primitive  root 
modulo  pa,  where  a  is  any  positive  integer. 
If 

7*=  1  mod  p, 

then  k  is  a  multiple  of  p  —  i,  since  7  is  a  primitive  root  modulo 
p.     Hence,  if 

7l=imodf, 

then  k  is  a  multiple  of  p  —  1 . 
Now,  write 

yt-^j+kp. 

Since  yp~1  —  1  is  not  divisible  by  ^>2,  it  follows  that  h  is  prime 
to  ^>.  If  we  raise  each  member  of  this  equation  to  the  power 
(3pa~2,  a>2J  we  have 

where  /  is  an  integer.     Then  if 

y>p«-2(P-i)^imod^ 

|8  must  be  divisible  by  p.  Therefore  the  exponent  of  the  lowest 
power  of  7  which  is  congruent  to  1  modulo  pa  is  divisible  by 
pa~l.  But  we  have  seen  that  this  exponent  is  also  divisible 
by  p  —  1.  Hence  the  exponent  of  7  modulo  pa  is  pa~1(p—i) 
since  (j>(pa)=pa~1(p  —  i).  That  is,  7  is  a*  primitive  root  mod- 
ulo pa. 

It  is  easy  to  see  that  no  two  numbers  of  the  set 

7,  72,  T3,   .    •    •   ,  7pa"1(p~1)  (A) 

are  congruent  modulo  pa;  for,  if  so,  7  would  belong  modulo  pa 
to  an  exponent  less  than  pa~1(p~i)  and  would  therefore  not 
be  a  primitive  root  modulo  pa.     Now  every  number  in  the  set 


70  THEORY   OF   NUMBERS 

(-4)  is  prime  to  pa;  their  number  is  <f>(pa)  =  pa~1(p  —  i).  Hence 
the  numbers  of  the  set  (A)  are  congruent  in  some  order  to  the 
numbers  of  the  set  (B) : 

0i,  02,  as,  ...  ,     0po-i(P_i),  (B) 

where  the  integers  (B)  are  the  positive  integers  less  than  pa 
and  prime  to  pa. 

But  any  number  of  the  set  (B)  is  a  solution  of  the  congruence 

/"^-"simodf.  (i) 

Further,  every  solution  of  this  congruence  is  prime  to  pa.  Hence 
the  integers  (B)  are  a  complete  set  of  solutions  of  (i).  Therefore 
the  integers  (A)  are  a  complete  set  of  solutions  of  (i).  But 
it  is  easy  to  see  that  an  integer  7*  of  the  set  (A)  is  or  is  not  a 
primitive  root  modulo  pa  according  as  k  is  or  is  not  prime  to 
pa~1(p—i).     Hence    the   number    of   primitive  roots   modulo 

The  results  thus  obtained  may  be  stated  as  follows: 
II.  If  p  is  any  odd  prime  number  and  a  is  any  positive  integer, 
then  there  exist  primitive  roots  modulo  pa  and  their  number  is 


§  37.  Primitive  Roots  Modulo  2pa,  p  an  Odd  Prime 

In  this  section  we  shall  prove  the  following  theorem : 
//  p  is  any  odd  prime  number  and  a  is  any  positive  integer, 
then  there  exist  primitive  roots  modulo  2pa  and  their  number  is 

*U(2f)}. 

Since  2pa  is  even  it  follows  that  every  primitive  root  modulo 
2pa  is  an  odd  number.  Any  odd  primitive  root  modulo  pa  is 
obviously  a  primitive  root  modulo  2pa.  Again,  if  7  is  an  even 
primitive  root  modulo  pa  then  y-\-pa  is  a  primitive  root  modulo 
2pa.  It  is  evident  that  these  two  classes  contain  (without 
repetition)  all  the  primitive  roots  modulo  2pa.  Hence  the 
theorem  follows  as  stated  above. 


primitive  roots  modulo  yyl  71 

§  38.  Recapitulation 

The  results  which  we  have  obtained  in  §§  34-37  inclusive 
may  be  gathered  into  the  following  theorem: 

In  order  that  there  shall  exist  primitive  roots  modulo  m,  it  is 
necessary  and  sufficient  that  m  shall  have  one  of  the  values 

m=i,  2,  4,  pa,  2pa 

where  p  is  an  odd  prime  and  a  is  a  positive  integer. 

If  m  has  one  of  these  values  then  the  number  of  primitive  roots 
modulo  m  is  <t>{4>{m) } . 


§39.  Primitive  X-roots 

In  the  preceding  sections  of  this  chapter  we  have  developed 
the  theory  of  primitive  roots  in  the  way  in  which  it  is  usually 
presented.  But  if  one  approaches  the  subject  from  a  more 
general  point  of  view  the  results  which  may  be  obtained  are 
more  general  and  at  the  same  time  more  elegant.  It  is  our 
purpose  in  this  section  to  develop  the  more  general  theory. 

We  have  seen  that  if  a  and  m  are  any  two  relatively  prime 
positive  integers,  then 

aHm)  =  j  m0(J  ^ 

Consequently  there  is  no  integer  belonging  modulo  m  to  an 
exponent  greater  than  \(m).  It  is  natural  to  enquire  if  there 
are  any  integers  a  which  belong  to  the  exponent  \(m).  It  turns 
out  that  the  question  is  to  be  answered  in  the  affirmative,  as 
we  shall  show.  Accordingly,  we  introduce  the  following  defini- 
tion: 

Definition.  If  aHm)  is  the  lowest  power  of  a  which  is 
congruent  to  1  modulo  m,  a  is  said  to  be  a  primitive  X-root 
modulo  m.  We  shall  also  say  that  it  is  a  primitive  X-root  of 
the  congruence  £x(m)  =  1  mod  m.  To  distinguish  we  may  speak 
of  the  usual  primitive  root  as  a  primitive  0-root  modulo  m. 


72  THEORY   OF   NUMBERS 

From  the  theory  of  primitive  0-roots  already  developed 
it  follows  that  primitive  X-roots  always  exist  when  m  is  a  power 
of  any  odd  prime,  and  also  when  m=  i,  2,  4;  for,  for  such  values 
of  m  we  have  \(m)  =  <f>(m). 

We  shall  next  show  that  primitive  X-roots  exist  when  m  =  2a, 
a>2,  .by  showing  that  5  is  such  a  root.  It  is  necessary  and  suf- 
ficient to  prove  that  5  belongs  modulo  2a  to  the  exponent 
2a_2  =  X(2a).  Let  d  be  the  exponent  to  which  5  belongs  modulo 
2a.  Then  from  theorem  II  of  §  32  it  follows  that  d  is  a  divisor 
of  2a-2  =  X(2a).  Hence  if  d  is  different  from  2a~2  it  is  2a"3 
or  is  a  divisor, of  2a~3.  Hence  if  we  can  show  that  fa"3  is  not 
congruent  to  1  modulo  2a  we  will  have  proved  that  5  belongs 
to  the  exponent  2a~2.     But,  clearly, 

52«-3  =  (I+22)2«-3=I+2«-l+/.2aj 

where  /  is  an  integer.     Hence 

52a_  ^1  mod  2a. 

Hence  5  belongs  modulo  2a  to  the  exponent  X(2a). 

By  means  of  these  special  results  we  are  now  in  position  to 
prove  readily  the  following  general  ,theorem  which  includes 
them  as  special  cases: 

I.  For  every  congruence  of  the  form 

xx(m)  =  i  mod  m 

a  solution  g  exists  which  is  a  primitive  \-root,  and  for  any  such 
solution  g  there  are  4>{\(m)\  primitive  roots  congruent  to  powers 

ofg-  ^   . 

If  any  primitive  X-root  g  exists,  gv  is  or  is  not  a  primitive 
X-root  according  as  v  is  or  is  not  prime  to  \{m) ;  and  therefore 
the  number  of  primitive  X-roots  which  are  congruent  to  powers 
of  any  such  root  g  is  <f>{\(m)  J . 

The  existence  of  a  primitive  X-root  in  every  case  may  easily 
be  shown  by  induction.  In  case  wis  a  power  of  a  prime  the 
theorem  has  already  been  established.  We  will  suppose  that 
it  is  true  when  m  is  the  product  of  powers  of  r  different  primes 


PRIMITIVE   ROOTS   MODULO   M  73 

and  show  that  it  is  true  when  m  is  the  product  of  powers  of 
r-f-i  different  primes;  from  this  will  follow  the  theorem  in 
general. 

Put     m  =  p1a'p2a2    •     •     •    prarparT^,       n  =  piaip2a2    .    .     .    prar, 

and  let  h  be  a  primitive  X-root  of 

xHn)  =  i  mod  n.  (i) 

Then 

h-\-ny 

is  a  form  of  the  same  root  if  y  is  an  integer. 
Likewise,  if  c  is  any  primitive  X-root  of 

^r+V^imod^1  (2) 

a  form  of  this  root  is 

where  z  is  any  integer. 

Now,  if  y  and  z  can  be  chosen  so  that 

h+ny  =  c+parr+11z 

the  number  in  either  member  of  this  equation  will  be  a  common 
primitive  X-root  of  congruences  (i)  and  (2);  that  is,  a  com- 
mon primitive  X-root  of  the  two  congruences  may  always  be 
obtained  provided  that  the  equation 


Pfi    .    .    .     pSry-p 


+  1 


has  always  a  solution  in  which  y  and  z  are  integers.  That  this 
equation  has  such  a  solution  follows  readily  from  theorem 
III  of  §  9;  for,  if  c  —  h  is  replaced  by  1,  the  new  equation  has  a 
solution  y,  z;  and  therefore  for  y  and  z  we  may  take  y  =  y(c  —  h), 
z=~z(c  —  h). 

Now  let  g  be  a  common  primitive  X-root  of  congruences 
(1)  and  (2)  and  write 

gv=i  mod  m, 


74  THEORY   OF    NUMBERS 

where  v  is  to  be  the  smallest  exponent  for  which  the  congruence 
is  true.  Since  g  is  a  primitive  X-root  of  (i)  v  is  a  multiple  of 
Mpiai  .  •  -'.prar).  Since  g  is  a  primitive  X-root  of  (2)  v  is  a 
multiple  of  X  (^^j1)  .  Hence  it  is  a  multiple  of  X(w).  But 
gx(m)  =  imodm;  therefore  v  =  \(m).  That  is,  g  is  a  primitive 
X-root  modulo  m. 

The  theorem  as  stated  now  follows  at  once  by  induction. 

There  is  nothing  in  the  preceding  argument  to  indicate 
that  the  primitive  X-roots  modulo  m  are  all  in  a  single  set 
obtained  by  taking  powers  of  some  root  g\  in  fact  it  is  not  in 
general  true  when  m  contains  more  than  one  prime  factor. 

By  taking  powers  of  a  primitive  X-root  g  modulo  m  one 
obtains  0{X(m)J  different  primitive  X-roots  modulo  m.  It  is 
evident  that  if  7  is  any  one  of  these  primitive  X-roots,  then  the 
same  set  is  obtained  again  by  taking  the  powers  of  7.  We  may 
say  then  that  the  set  thus  obtained  is  the  set  belonging  to  g. 

II.  7/  X(w)>2  the  product  of  the  <f>{\(m)}  primitive  \-roots 
in  the  set  belonging  to  any  primitive  \-root  g  is  congruent  to  1 
modulo  m. 

These  primitive  X-roots  are 

8>  8  )  8  >  •  •  •  5  8 
where 

I,   Ci,  C2)   •   *    •    ,   Cft 

are  the  integers  less  than  \(m)  and  prime  to  X(w).  If  any  one 
of  these  is  c  another  is  \(m)—c,  since  \(m)>2.     Hence 

1+C1+C2+  •  •  •   +cM  =  o  mod  X(ra). 
Therefore 

gi+Cl+C2+...  +cME=I  mod  w. 

From  this  the  theorem  follows. 

Corollary.  The  product  of  all  the  primitive  \-roots  modulo 
m  is  congruent  to  1  modulo  m  when  \{m)  >  2. 


PRIMITIVE    ROOTS   MODULO   M  75 


EXERCISES 


i.  If  *i  is  the  largest  value  of  x  satisfying  the  equation  \(x)=a,  where  a  is 
a  given  integer,  then  any  solution  x2  of  the  equation  is  a  factor  of  X\. 

2*.  Obtain  an  effective  rule  for  solving  the  equation  \{x)—a. 

3*.  Obtain  an  effective  rule  for  solving  the  equation  <f>(x)=a. 

4.  A  necessary  and  sufficient  condition  that  ap_1  =  i  mod  P  for  every  in- 
teger a  prime  to  P  is  that  P  =  i  mod  X(P). 

5.  If  ap~l  =  i  mod  P  for  every  a  prime  to  P,  then  (1)  P  does  not  contain  a 
square  factor  other  than  1,  (2)  P  either  is  prime  or  contains  at  least  three  dif- 
ferent prime  factors. 

6.  Let  p  be  a  prime  number.  If  a  is  a  root  of  the  congruence  xd=  1  mod  p  and 
a  is  a  root  of  the  congruence  xs  =  i  mod  p,  then  aa  is  a  root  of  the  congruence 
xds  =  i  mod  p.  If  a  is  a  primitive  root  of  the  first  congruence  and  a  of  the  second 
and  if  d  and  5  are  relatively  prime,  then  aa  is  a  primitive  root  of  the  congruence 
xdb  =  i  mod  p. 


CHAPTER  VI 
OTHER  TOPICS 

§  40.  Introduction 

The  theory  of  numbers  is  a  vast  discipline  and  no  single 
volume  can  adequately  treat  of  it  in  all  of  its  phases.  A  short 
book  can  serve  only  as  an  introduction;  but  where  the  field 
is  so  vast  such  an  introduction  is  much  needed.  That  is  the 
end  which  the  present  volume  is  intended  to  serve;  and  it 
will  best  accomplish  this  end  if,  in  addition  to  the  detailed  theory 
already  developed,  some  account  is  given  of  the  various  direc- 
tions in  which  the  matter  might  be  carried  further. 

To  do  even  this  properly  it  is  necessary  to  limit  the  number 
of  subjects  considered.  Consequently  we  shall  at  once  lay 
aside  many  topics  of  interest  which  would  find  a  place  in  an 
exhaustive  treatise.  We  shall  say  nothing,  for  instance,  about 
the  vast  domain  of  algebraic  numbers,  even  though  this  is  one 
of  the  most  fascinating  subjects  in  the  whole  field  of  mathe- 
matics. Consequently,  we  shall  not  refer  to  any  of  the  exten- 
sive theory  connected  with  the  division  of  the  circle  into  equal, 
parts.  Again,  we  shall  leave  unmentioned  many  topics  con- 
nected with  the  theory  of  positive  integers;  such,  for  instance, 
is  the  frequency  of  prime  numbers  in  the  ordered  system  of 
integers — a  subject  which  contains  in  itself  an  extensive  and 
elegant  theory. 

In  §§  41-44  we  shall  speak  briefly  of  each  of  the  following 
topics:  theory  of  quadratic  residues,  Galois  imaginaries,  arith- 
metic forms,  analytical  theory  of  numbers.  Each  of  these  alone 
would  require  a  considerable  volume  for  its  proper  development. 
All  that  we  can  do  is  to  indicate  the  nature  of  the  problem  in 
each  case  and  in  some  cases  to  give  a  few  of  the  fundamental 
results. 

76 


OTHER   TOPICS  77 

In  the  remaining  three  sections  we  shall  give  a  brief  intro- 
duction to  the  theory  of  Diophantine  equations,  developing 
some  of  the  more  elementary  properties  of  certain  special 
cases.  We  shall  carry  this  far  enough  to  indicate  the  nature 
of  the  problem  connected  with  the  now  famous  Last  Theorem 
of  Fermat.  The  earlier  sections  of  this  chapter  are  not  required 
as  a  preliminary  to  reading  this  latter  part. 


§41.  Theory  of  Quadratic  Residues 

Let  a  and  m  he  any  two  relatively  prime  integers.  In  §  31 
we  agreed  to  say  that  a  is  a  quadratic  residue  modulo  m  or  a 
quadratic  non-residue  modulo  m  according  as  the  congruence 

x2  =  a  mod  m 

has  or  has  not  a  solution.  We  saw  that  if  m  is  chosen  equal 
to  an  odd  prime  number  p,  then  a  is  a  quadratic  residue  modulo 
p  or  a  quadratic  non-residue  modulo  p  according  as 

aKp-i)  =  !      or     fli(P-D=_imo(J|,, 

This  is  known  as  Euler's  criterion. 

It  is  convenient  to  employ  the  Legendre  symbol 


-;) 


to  denote  the  quadratic  character  of  a  with  respect  to  p.  This 
symbol  is  to  have  the  value  +1  or"  the  value  —  1  according 
as  a  is  a  quadratic  residue  modulo  p  or  a  quadratic  non-residue 
modulo  p.  We  shall  now  derive  some  of  the  fundamental  prop- 
erties of  this  symbol,  understanding  always  that  the  numbers 
in  the  numerator  and  the  denominator  are  relatively  prime. 

From  the  definition  of  quadratic  residues  and  non-residues 
it  is  obvious  that 


($)-( 


if  a  =  b  mod  p.  (1) 

Pi 


78  THEORY   OF   NUMBERS 

It  is  easy  to  prove  in  general  that 


(?)(!)-(t) 


(2) 


This  comes  readily  from  Euler's  criterion.     We  have  *to  con- 
sider the  three  cases 

(I)--  ©-+«  ©-+■.  ©-< 

f)— •  (|)—- 

The  method  will  be  sufficiently  illustrated  by  the  treatment 
of  the  last  case.     Here  we  have 

aiiP-D^^j  modp,     ^Kp-D=_t  mod/>. 

Multiplying  these  two  congruences  together  member  by  member 
we  have 

O^-^imOd^,    '. 


whence 


(f)--m- 


as  was  to  be  proved. 

If  m  is  any  number  prime  to  p  and  we  write  m  as  the  product 
of  factors 

w  =  e.2*V<z'Y"  •  •  • 

where  q',  q",  q/;;,  ...  are  odd  primes,  a  is  zero  or  a  positive 
integer  and  e  is  + 1  or  —  1  according  as  m  is  positive  or  negative, 
we  have 

"Hmmm  •  •  •  ■    « 

as  one  shows  easily  by  repeated  application  of  relation  (2). 
Obviously, 


©- 


OTHER   TOPICS  79 

Hence,  it  follows  from  (3)  that  we  can  readily  determine  the 
quadratic  character  of  m  with  respect  to  the  odd  prime  p,  that 
is,  the  value  of 

■      ••        ©:,.        • 

provided  that  we  know  the  value  of  each  of  the  expressions 

where  q  is  an  odd  prime. 

The  first  of  these  can  be  evaluated  at  once  by  means  of 
Euler's  criterion;     for,  we  have 


(f)-< 


and  hence 

'  =(_I)KP-D. 


(f)-<- 


Thus  we  have  the  following  result:  The  number  —  1  is  a  quad- 
ratic residue  of  every  prime  number  of  the  form  4&  +  1  and 
a  quadratic  non-residue  of  every  prime  number  of  the  form 

4^+3- 

The  value  of  the  second  symbol  in  (4)  is  given  by  the  formula 


(?) 


=  (_I)i(P-i). 


The  theorem  contained  in  this  equation  may  be  stated  in  the 
following  words:  The  number  2  is  a  quadratic  residue  of  every 
prime  number  of  either  of  the  forms  8&  +  1,  8^  +  7;  it  is  a  quad- 
ratic non-residue  of  every  prime  number  of  either  of  the  forms 
8£+3,  8£  +  5. 

The  proof  of  this  result  is  not  so  immediate  as  that  of  the 
preceding  one.  To  evaluate  the  third  expression  in  (4)  is  still 
more  difficult.  We  shall  omit  the  demonstration  in  both  of 
these  cases.     For  the  latter  we  have  the  very  elegant  relation 


©©=<-> 


i(P-i)(a-i) 


80  THEORY   OF   NUMBERS 

This  equation  states  the  law  which  connects  the  quadratic 
character  of  q  with  respect  to  p  with  the  quadratic  character 
of  p  with  respect  to  q.  It  is  known  as  the  Law  of  Quadratic 
Reciprocity.  About  fifty  proofs  of  it  have  been  given.  Its 
history  has  been  a  very  interesting  one;  see  Bachmann's 
Niedere  Zahlentheorie,  Teil  I,  pp.  180-318,  especially  pp. 
200-206. 

For  a  further  account  of  this  beautiful  and  interesting 
subject  we  refer  the  reader  to  Bachmann,  loc.  cit.,  and  to  the 
memoirs  to  which  this  author  gives  reference. 

§  42.  Galois  Imaginaries 

If  one  is  working  in  the  domain  of  real  numbers  the  equation 

#2+i=o 

has  no  solution;  for  there  is  no  real  number  whose  square  is 
—  1.  If,  however,  one  enlarges  the  "  number  system"  so  as 
to  include  not  only  all  real  numbers  but  all  complex  numbers 
as  well,  then  it  is  true  that  every  algebraic  equation  has  a  root. 
It  is  on  account  of  the  existence  of  this  theorem  for  the  enlarged 
domain  that  much  of  the  general  theory  of  algebra  takes  the 
elegant  form  in  which  we  know  it. 

The  question  naturally  arises  as  to  whether  we  can  make  a 
similar  extension  in  the  case  of  congruences.     The  congruence 

#2  =  3  mod  5 

has  no  solution,  if  we  employ  the  term  solution  in  the  sense  in 
which  we  have  so  far  used  it.  But  we  may  if  we  choose  intro- 
duce an  imaginary  quantity,  or  mark,  j  such  that 

jf2  =  3  mod  5, 

just  as  in  connection  with  trie  equation  #2+i=o  we  would 

introduce  the  symbol  i  having  the  property  expressed  by  the 

equation 

;2  =  _  T 


OTHER   TOPICS  81 

It  is  found  to  be  possible  to  introduce  in  this  way  a  general 
set  of  imaginaries  satisfying  congruences  with  prime  moduli; 
and  the  new  quantities  or  marks  have  the  property  of  combining 
according  to  the  laws  of  algebra. 

The  quantities  so  introduced  are  called  Galois  imaginaries. 

We  cannot  go  into  a  development  of  the  important  theory 
which  is  introduced  in  this  way.  We  shall  be  content  with 
indicating  two  directions  in  which  it  leads. 

In  the  first  place  there  is  the  general  Galois  field  theory 
which  is  of  fundamental  importance  in  the  study  of  certain 
finite  groups.  It  may  be  developed  from  the  point  of  view 
indicated  here.  An  excellent  exposition,  along  somewhat 
different  lines,  is  to  be  found  in  Dickson's  Linear  Groups  with 
an  Exposition  of  the  Galois  Field  Theory. 

Again,  the  whole  matter  may  be  looked  upon  from  the  geo- 
metric point  of  view.  In  this  way  we  are  led  to  the  general 
theory  of  finite  geometries,  that  is,  geometries  in  which  there 
is  only  a  finite  number  of  points.  For  a  development  of  the 
ideas  which  arise  here  see  Veblen  and  Young's  Projective 
Geometry  and  the  memoir  by  Veblen  and  Bussey  in  the  Trans- 
actions of  the  American  Mathematical  Society,  vol.  7,  pp. 
241-259. 

§  43.  Arithmetic  Forms 

The  simplest  arithmetic  form  is  ax+b  where  a  and  b  are 
fixed  integers  different  from  zero  and  x  is  a  variable  integer. 
By  varying  x  in  this  case  we  have  the  terms  of  an  arithmetic 
progression.  We  have  already  referred  to  Dirichlet's  cele- 
brated theorem  which  asserts  that  the  form  ax+b  has  an  infinite 
number  of  prime  values  if  only  a  and  b  are  relatively  prime. 
This  is  an  illustration  of  one  type  of  theorem  connected  with 
arithmetic  forms  in  general,  namely,  those  in  which  it  is  asserted 
that  numbers  of  a  given  form  have  in  addition  a  given  property. 

Another  type  of  theorem  is  illustrated  by  a  result  stated 
in  §41,  provided  that  we  look  at  that  result  in  the  proper 
way.     We  saw  that  the  number  2  is  a  quadratic  residue  of 


82  THEORY   OF   NUMBERS 

every  prime  of  either  of  the  forms  Sk  + 1  and  &k  +  j  and  a  quad- 
ratic non-residue  of  every  prime  of  either  of  the  forms  8^+3 
and  8^  +  5.  We  may  state  that  result  as  follows:  A  given 
prime  number  of  either  of  the  forms  Sk-\-i  and  Sk  +  7  is  a  divisor 
of  some  number  of  the  form  x2  —  2,  where  x  is  an  integer;  no 
prime  number  of  either  of  the  forms  8^+3  and  8^  +  5  is  a 
divisor  of  a  number  of  the  form  x2  —  2,  where  x  is  an  integer. 

The  result  just  stated  is  a  theorem  in  a  discipline  of  vast 
extent,  namely,  the  theory  of  quadratic  forms.  Here  a  large 
number  of  questions  arise  among  which  are  the  following: 
What  numbers  can  be  represented  in  a  given  form?  What  is 
the  character  of  the  divisors  of  a  given  form?  As  a  special 
case  of  the  first  we  have  the  question  as  to  what  numbers  can 
be  represented  as  the  sum  of  three  squares.  To  this  category 
belong  also  the  following  two  theorems:  Every  positive  integer 
is  the  sum  of  four  squares  of  integers;  every  prime  number  of 
the  form  4^+1  may  be  represented  (and  in  only  one  way)  as 
the  sum  of  two  squares. 

For  an  extended  development  of  the  theory  of  quadratic 
forms  we  refer  the  reader  to  Bachmann's  Arithmetik  der  Quad- 
ratischen  Formen  of  which  the  first  part  has  appeared  in  a 
volume  of  nearly  seven  hundred  pages. 

It  is  clear  that  one  may  further  extend  the  theory  of  arith- 
metic forms  by  investigating  the  properties  of  those  of  the  third 
and  higher  degrees.  Naturally  the  development  of  this  subject 
has  not  been  carried  so  far  as  that  of  quadratic  forms;  but 
there  is  a  considerable  number  of  memoirs  devoted  to  various 
parts  of  this  extensive  field,  and  especially  to  the  consideration 
of  various  special  forms. 

Probably  the  most  interesting  of  these  special  forms  are  the 
following : 

n nn 

niQti        **         P    n-l_l_    »-2/d_i_  1    pn-1 

a  +j8  ,      ^=«       +<*       P  +   •   •   •    +P       > 

a  —  p 

where  a  and  j8  are  relatively  prime  integers,  or,  more  generally, 
where  a  and  0  are  the  roots  of  the  quadratic  equation 
x2— ux+v  —  o  where  u  and  v  are  relatively  prime  integers.     A 


..  &<ftxA+*    \     „    t4m^^^^.  83 

development  of  the  theory  of  these  forms  has  been  given  by 
the  present  author  in  a  memoir  published  in  19 13  in  the  Annals 
of  Mathematics,  vol.  13,  pp.  30-70. 


§  44.  Analytical  Theory  of  Numbers 
Let  us  consider  the  function 


P(x)=- 

I 

\x\ 

Sp<l. 

00                        > 

n  (i-*2t) 

fc=0 

It 

is  clear  that 

we  have 

p(*)= 

00 

■  n 

1 

)  I—  X2* 

=  n  (i+x2t 

k=0 

=  2  G(s)xs, 

8=0 

+X2 

■2t+x3- 

where  G(o)  =  1  and  G(s)  (for  5  greater  than  o)  is  the  number  of 
ways  in  which  the  positive  integer  s  may  be  separated  into  like 
or  distinct  summands  each  of  which  is  a  power  of  2. 
We  have  readily 

(i-xy2G(s)xs  =  (i-x)P(x)=P(x2)=  I  G(s)x2S; 

8=0  s=0 

whence 

G(2s+i)  =G(2s)  =G(2s-i)+G(s),  (A) 

as  one  readily  verifies  by  equating  coefficients  of  like  powers 
of  x.     From  this  we  have  in  particular 

G(o)  =  i,    G(i)  =  i,    G(2)  =  2,    G(3)  =  2, 
G(4)=4,    G(5)=4,    G(6)=6,    G(7)=6. 

Thus  in  (-4)  we  have  recurrence  relations  by  means  of  which 
we  may  readily  reckon  out  the  values  of  the  number  theoretic 
function  G(s).  Thus  we  may  determine  the  number  of  ways  in 
which  a  given  positive  integer  5  may  be  represented  as  a  sum 
of  powers  of  2. 


84  THEORY   OF   NUMBERS 

We  have  given  this  example  as  an  elementary  illustration 
of  the  analytical  theory  of  numbers,  that  is,  of  that  part  of  the 
theory  of  numbers  in  which  one  employs  (as  above)  the  theory 
of  a  continuous  variable  or  some  analogous  theory  in  order  to 
derive  properties  of  sets  of  integers.  This  general  subject 
has  been  developed  in  several  directions.  For  a  systematic 
account  of  it  the  reader  is  referred  to  Bachmann's  Analytische 
Zahlentheorie. 


§  45.  Diophantine  Equations 

If  f(x,  y,z,  .  .  .  )  is  a  polynomial  in  the  variables  x,  y,  z,  .  .  . 
with  integral  coefficients,  then  the  equation 

f(x,  y,  z,  .  .  .  )=o 

is  called  a  Diophantine  equation  when  we  look  at  it  from  the 
point  of  view  of  determining  the  integers  (or  the  positive  in- 
tegers) x,  y,  z,  .  .  .  which  satisfy  it.  Similarly,  if  we  have 
several  such  functions  /*(#,  y,  z,  .  .  .  ),  in  number  less  than 
the  number  of  variables  x,  y,  z,  .  .  .  ,  then  the  set  of  equations 

ft(x,  y,z,  .  .  .)=o,     i  =  i,  2,    .  .  .  , 

is  said  to  be  a  Diophantine  system  of  equations.  Any  set  of 
integers  x,  y,  z,  .  .  .  which  satisfies  the  equation  [system] 
is  said  to  be  a  solution  of  the  equation  [system]. 

We  may  likewise  define  Diophantine  inequalities  by  replac- 
ing the  sign  of  equality  above  by  the  sign  of  inequality.  But 
little  has  been  done  toward  developing  a  theory  of  Diophantine 
inequalities.  Even  for  Diophantine  equations  the  theory  is 
in  a  rather  fragmentary  state. 

In  the  next  two  sections  we  shall  illustrate  the  nature  of 
the  ideas  and  the  methods  of  the  theory  of  Dipohantine  equa- 
tions by  developing  some,  of  the  results  for  two  important 
special  cases. 


OTHER   TOPICS  85 


§  46.  Pythagorean  Triangles 

Definitions.  If  three  positive  integers  x,  y,  z  satisfy 
the  relation 

x2+y2  =  z2  (1) 

they  are  said  to  form  a  Pythagorean  triangle  or  a  numerical 
right  triangle;  z  is  called  the  hypotenuse  of  the  triangle  and  x 
and  y  are  called  its  legs.  The  area  of  the  triangle  is  said  to  be 
\xy. 

We  shall  determine  the  general  form  of  the  integers  x,  y,  z, 
such  that  equation  (i)  may  be  satisfied.  Let  us  denote  by  v 
the  greatest  common  divisor  of  x  and  y  in  a  particular  solution 
of  (1).     Then  v  is  a  divisor  of  z  and  we  may  write 

x  =  vu,     y  =  w,     z  =  vw. 

Substituting  these  values  in  (i)  and  reducing  we  have 

u2+v2=w2,  (2) 

where  u,  v,  w  are  obviously  prime  each  to  each,  since  u  and  v 
have  the  greatest  common  divisor  i. 

Now  an  odd  square  is  of  the  form  4&  +  1.  Hence  the  sum 
of  two  odd  squares  is  divisible  by  2  but  not  by  4;  and  therefore 
the  sum  of  two  odd  squares  cannot  be  a  square.  Hence  one 
of  the  numbers  u,  v  is  even.  Suppose  that  u  is  even  and  write 
equation  (2)  in  the  form 

u2  =  (w— v)(w+v).  (3) 

Every  common  divisor  oi  w—v  and  w+v  is  a  divisor  of  their 
difference  2V.  Therefore,  since  w  and  v  are  relatively  prime, 
it  follows  that  2  is  the  greatest  common  divisor  oi  w—v  and 
w+v.  Then  from  (3)  we  see  that  each  of  these  numbers  is 
twice  a  square,  so  that  we  may  write 

w— V  =  2b2,     w+v  =  2a2 


86  THEORY    OF   NUMBERS 

where  a  and  b  are  relatively  prime  integers.     From  these  two 
equations  and  equation  (3)  we  have 

w  =  a2+b2,     v  =  a2  —  b2,     u  =  2ab.  (4) 

Since  u  and  v  are  relatively  prime  it  is  evident  that  one  of  the 
numbers  a,  b  is  even  and  the  other  odd. 

The  forms  of  u,  v,  w  given  in  (4)  are  necessary  in  order  that 
(2)  may  be  satisfied.  A  direct  substitution  in  (2)  shows  that 
this  equation  is  indeed  satisfied  by  these  values.  Hence  we 
have  in  (4)  the  general  solution  of  (2)  where  u  is  restricted  to 
be  even.  A  similar  solution  would  be  obtained  if  v  were  re- 
stricted to  be  even.     Therefore  the  general  solution  of  (1)   is 

x  =  2vab,     y  =  v(a2  —  b2),     z  =  v(a2+b2) 
and 

x  =  v(a2  —  b2),     y  =  2vab,     z  =  v(a2+b2) 

where  a,  b,  v  are  arbitrary  integers  except  that  a  and  b  are  rela- 
tively prime  and  one  of  them  is  even  and  the  other  odd. 

By  means  of  this  general  solution  of  (1)  we  shall  now  prove 
the  following  theorem: 

I.  There  do  not  exist  integers  m,  n,  p,  q,  all  different  from 
zero,  such  that 

q2-\-n2  =  m2,     m2-\-n2=p2.  (5) 

It  is  obvious  that  an  equivalent  theorem  is  the  following: 

II.  There  do  not  exist  integers  m,  n,  p}  q,  all  different  from 
zero  such  that 

p2+q2  =  2m2,     p2—q2  —  2n2.  (6) 

Obviously,  we  may  without  loss  of  generality  take  m,  n, 
p,  q  to  be  positive;  and  this  we  do. 

The  method  of  proof  is  to  assume  the  existence  of  integers 
satisfying  equations  (5)  and  (6)  and  to  show  that  we  are  thus 
led  to  a  contradiction.  The  argument  we  give  is  an  illustra- 
tion of  Fermat's  famous  method  of  "  infinite  descent.' ' 

If  any  two  of  the  numbers  p,  q,  m,  n  have  a  common  prime 
factor  /,  it  follows  at  once  from  (5)  and  (6)  that  all  four  of  them 


OTHER   TOPICS  87 

have  this  factor.  For,  consider  an  equation  in  (5)  or  in  (6) 
in  which  these  two  numbers  occur;  this  equation  contains  a 
third  number,  and  it  is  readily  seen  that  this  third  number  is 
divisible  by  /.  Then  from  one  of  the  equations  containing  the 
fourth  number  it  follows  that  this  fourth  number  is  divisible 
by  /.  Now  let  us  divide  each  equation  of  system  (6)  through 
by  t2;  the  resulting  system  is  of  the  same  form  as  (6).  If 
any  two  numbers  in  this  resulting  system  have  a  common  prime 
factor  h,  we  may  divide  through  by  h2;  and  so  on.  Hence  if 
a  pair  of  simultaneous  equations  (6)  exists  then  there  exists  a 
pair  of  equations  of  the  same  form  in  which  no  two  of  the  num- 
bers m,  n,  p,  q  have  a  common  factor  other  than  unity.  Let 
this  system  of  equations  be 

pi2+qi2  =  2Mi2,     pi2-qi2  =  2tii2.  (7) 

From  the  first  equation  in  (7)  it  follows  that  pi  and  q\  are 
both  even  or  both  odd;  and,  since  they  are  relatively  prime, 
it  follows  that  they  are  both  odd.  Evidently  pi>qi.  Then 
we  may  write 

pi=qi  +  2<x, 

where  a  is  a  positive  integer.  If  we  substitute  this  value  of 
pi  in  the  first  equation  of  (7),  the  result  may  readily  be  put  in 

the  form 

(qi+a)2+a2  =  mi2.  (8) 

Since  qi  and  mi  have  no  common  prime  factor  it  is  easy  to  see 
from  this  equation  that  a  is  prime  to  both  qi  and  mi,  and  hence 
that  no  two  of  the  numbers  qi  -\-a,  a,  mi  have  a  common  factor. 

Now  we  have  seen  that  if  a,  b,  c  are  positive  integers  no  two 
of  which  have  a  common  prime  factor,  while 

a2+b2  =  c2, 

then  there  exist  relatively  prime  integers  r  and  s}  r>s,  such 

that 

c  =  r2-\-s2,     a  =  2rs,     b  =  r2—s2 

or 

c  =  r2+s2,     a  =  r2  —  s2,     b  =  2rs. 


88  THEORY   OF   NUMBERS 

Hence  from  (8)  we  see  that  we  may  write 

qi-\-a  =  2rs,    a=r2—s2  (9) 

or 

qi+a  =  r2  —  s2,     a  =  2TS.  (10) 

In  either  case  we  have 

pi2-qi2  =  (pi-qi)(pi+qi)  =  2a-2(q1+a)=Srs(r2-s2). 

If  we  substitute  in  the  second  equation  of  (7)  and  divide  by  2 

we  have 

4rs(r2— s2)=n\2. 

From  this  equation  and  the  fact  that  r  and  5  are  relatively 
prime  it  follows  at  once  that  r,  s,  r2  —  s2  are  all  square  numbers; 
say, 

r  =  u2,    s  =  v2,    r2—s2  =  w2. 

Now  r—s  and  r+s  can  have  no  common  factor  other  than 
1  or  2 ;  hence  from 

w2  =  (y2  _  S2^  —  (y  _  5)  (y  +s^  =  (u2  _  v2^  (u2  _^_v2^ 

we  see  that  either 

U2  ~\-V2  =  2Wi2 ,      U2  —  V2  =  2W22  (11) 

or 

U2~\-V2=Wi2,      U2—V2  =  W22. 

And  if  it  is  the  latter  case  which  arises,  then 

Wi2  -\-W22  =  2U2 ,       Wi2—  W22  =  2V2.  (12) 

Hence,  assuming  equations  of  the  form  (6)  we  are  led  either  to 
equations  (n)  or  to  equations  (12);  that  is,  we  are  led  to  new 
equations  of  the  form  with  which  we  started.  Let  us  write 
the  equations  thus: 

p22+q22  =  2tn22,    p22-q22  =  2n22;  (13) 

that  is,  system  (13)  is  identical  with  that  one  of  systems  (11), 
(12)  which  actually  arises. 


OTHER   TOPICS  89 

Now  from  (9)  and  (10)  and  the  relations  pi=qi  +  2a,  r>s, 
we  see  that 

pi  =  2rs+r2-s2>2s2+r2-s2  =  r2+s2  =  u*+v*. 

Hence  u<p\.     Also, 

wi2  ^w2  =  r+s  <r2+s2. 

Hence  wi<pi.  Since  u  and  w\  are  both  less  than  pi  it  follows 
that  p2  is  less  than  pi.  Hence,  obviously,  p2<p-  Moreover, 
it  is  clear  that  all  the  numbers  ^2,  £2,  W2,  n^  are  different  from 
zero. 

'  From  these  results  we  have  the  following  conclusion :  If 
we  assume  a  system  of  the  form  (6)  we  are  led  to  a  new  system 
(13)  of  the  same  form;  and  in  the  new  system  p2  is  less  than  p. 

Now  if  we  start  with  (13)  and  carry  out  a  similar  argument 
we  shall  be  led  to  a  new  system 

P32 +qs2  =  2  w32,     ps2  -  qz2  =  2fl32, 

with  the  relation  pz<p2\  starting  from  this  last  system  we  shall 
be  led  to  a  new  one  of  the  same  form,  with  a  similar  relation  of 
inequality;  and  so  on  ad  infinitum.  But,  .since  there  is  only 
a  finite  number  of  positive  integers  less  than  the  given  positive 
integer  p  this  is  impossible.  We  are  thus  led  to  a  contradic- 
tion; whence  we  conclude  at  once  to  the  truth  of  II  and  like- 
wise of  I. 

By  means  of  theorems  I  and  II  we  may  readily  prove  the 
following  theorem: 

III.  The  area  of  a  numerical  right  triangle  is  never  a  square 
number. 

Let  the  sides  and  hypotenuse  of  a  numerical  right  triangle 
be  u,  v,  w,  respectively.  The  area  of  this  triangle  is  \ui).  If 
we  assume  this  to  be  a  square  number  t2  we  shall  have  the 
following  simultaneous  Diophantine  equations 

U2+V2=W2,       UV  =  2t2.  (14) 


90  THEORY   OF   NUMBERS 

We  shall  prove  our  theorem  by  showing  that  the  assumption 
of  such  a  system  leads  to  a  contradiction. 

If  any  two  of  the  numbers  u,  v,  w  have  a  common  prime 
factor  p  then  the  remaining  one  also  has  this  factor,  as  one 
sees  readily  from  the  first  equation  in  (14).  From  the  second 
equation  in  (14)  it  follows  that  t  also  has  the  same  factor.  Then 
if  we  put  u  =  pui,  v  —  pvij  w  =  pwi,  t  =  ph,  we  have 


Ui2  +V\2  =  Wi2,      U1V1  =  2ti2, 

a  system  of  the  same  form  as  (14).  It  is  clear  that  we  may 
start  with  this  new  system  and  proceed  in  the  same  manner  as 
before,  and  so  on,  until  we  arrive  at  a  system 

U2  +  V2=W2,       UV  =  2~t2,  (15) 

where  u,  v,  w  are  prime  each  to  each. 

Now  the  general  solution  of  the  first  equation  (15)  may  be 
written  in  one  of  the  forms 

u  =  2ab,     v  =  a2  —  b2,     w  =  a2-\-b2, 
u  =  a2b2,     v  =  2ab,         w  =  a2+b2. 

Then  from  the  second  equation  in  (15)  we  have 

}2  =  ab(a2-b2)=ab(a-b)(a+b). 

It  is  easy  to  see  that  no  two  of  the  numbers  a,  b,  a  —  b,  a-\-b 
in  the  last  member  of  this  equation  have  a  common  factor;  for, 
if  so,  u  and  v  would  have  a  common  factor,  contrary  to  hypoth- 
esis. Hence  each  of  these  four  numbers  is  a  square.  That  is, 
we  have  equations  of  the  form 

a  =  tn2,     b  =  n2,     a+b  =  p2,     a  —  b=q2\ 
whence  m2—n2  =  q2,    m2-\-n2=  p2. 

But,  according  to  theorem  I,  no  such  system  of  equations  can 
exist.  That  is,  the  assumption  of  equations  (14)  leads  to  a 
contradiction.     Hence  the  theorem  follows  as  stated  above. 


OTHER   TOPICS  91 


§47.    The  Equation  xn+yn  =  zn. 

The  following  theorem,  which  is  commonly  known  as  Fer- 
mat's  Last  Theorem,  was  stated  without  proof  by  Fermat 
in  the  seventeenth  century: 

//  n  is  an  integer  greater  than  2  there  do  not  exist  integers 
x,  y,  z,  all  different  from  zero,  such  that 

xn+yn  =  z\  (1) 

No  general  proof  of  this  theorem  has  yet  been  given.  For 
various  special  values  of  n  the  proof  has  been  found;  in  par- 
ticular, for  every  value  of  n  not  greater  than  100. 

In  the  study  of  equation  (1)  it  is  convenient  to  make  some 
preliminary  reductions.  If  there  exists  any  particular  solution 
of  (1)  there  exists  also  a  solution  in  which  x,  y,  z  are  prime 
each  to  each,  as  one  may  show  readily  by  the  method  employed 
in  the  first  part  of  §  46.  Hence  in  proving  the  impossibility 
of  equation  (1)  it  is  sufficient  to  treat  only  the  case  in  which 
x,  y,  z  are  prime  each  to  each. 

Again,  since  n  is  greater  than  2  it  must  contain  the  factor 
4  or  an  odd  prime  factor  p.  If  n  contains  the  factor  p  we  write 
n  =  mp,  whence  we  have 

(xmy+(ym)p=(zmy. 

If  n  contains  the  factor  4  we  write  n  =  4m,  whence  we  have 

(xmy+(ymy={zmy. 

From  this  we  see  that  in  order  to  prove  the  impossibility  of 
(1)  in  general  it  is  sufficient  to  prove  it  for  the  special  cases  when 
n  is  4  and  when  n  is  an  odd  prime  p.  For  the  latter  case  the 
proof  has  not  been  found.  For  the  former  case  we  give  a 
proof  below.     The  theorem  may  be  stated  as  follows: 

I.  There  are  no  integers  x,  y,  z,  all  different  from  zero,  such 
that 


92  THEORY   OF   NUMBERS 

This  is  obviously  a  special  case  of  the  more  general  theorem : 
II.  There  are  no  integers  p,  q,  a,  all  different  from  zero,  such 
that 

P4-q4=a2.  (2) 

The  latter  theorem  is  readily  proved  by  means  of  theorem 
III  of  §46.  For,  if  we  assume  an  equation  of  the  form  (2), 
we  have 

(p*-q*)p2q2  =  PY<*2-  (3) 

But,  obviously, 

(2^Y)2+(^4-g4)2  =  (^+g4)2.  (4) 

Now,  from  (3)  we  see  that  the  numerical  right  triangle  deter- 
mined by  (4)  has  its  area  p2q2(p4:  —  q4)  equal  to  the  square  num- 
ber p2q2a2.  But  this  is  impossible.  Hence  no  equation  of  the 
form  (2)  exists. 

EXERCISES 

1.  Show  that  the  equation  a4+4/34  =  72  is  impossible  in  integers  a,  /3,  7,  all 
of  which  are  different  from  zero. 

2.  Show  that  the  system  p2—q2  =  km2,  p2-\-q2  =  kn2  is  impossible  in  integers 
p,  q,  k,  m,  n,  all  of  which  are  different  from  zero. 

3*.  Show  that  neither  of  the  equations  m4— 4tii=dbt2  is  possible  in  integers 
m,  n,  t,  all  of  which  are  different  from  zero. 

4*.  Prove  that  the  area  of  a  numerical  right  triangle  is  not  twice  a  square 
number. 

5*.  Prove  that  the  equation  mi-\-n*=a2  is  not  possible  in  integers  m,  n,  a  all 
of  which  are  different  from  zero. 

6*.  In  the  numerical  right  triangle  a2+b2=c2,  not  more  than  one  of  the  num- 
bers a,  b,  c  is  a  square. 

7.  Prove  that  the  equation  x  -\-y  =z2  implies  an  equation  of  the  form 
m  +n=2*     t  . 

8.  Find  the  general  solution  in  integers  of  the  equation  x2+2y2=t2. 

9.  Find  the  general  solution  in  integers  of  the  equation  x2+y*=zA. 

10.  Obtain  solutions  of  each  of  the  following  Diophantine  equations: 

xz-\-y3+zz  =  2t3, 

x*+y4+4z*  =  t\ 

X*+V4+Z4=2t*. 


INDEX 


Algebraic  numbers,  76. 
Analytical  theory  of  numbers,  83-84. 
Arithmetic  forms,  81-83. 
Arithmetic  progression,  13. 

Bachmann,  80,  82,  84. 
Bussey,  81. 

Carmichael,  83. 
Circle,  Division  of,  76. 
Common  divisors,  9,  18-20,  21. 

multiples,  9,  20-21. 
Composite  numbers,  10.  fc 

Congruences,  37-46. 

Linear,  43-46,  56. 

Solution  by  trial,  39-40. 

with  prime  modulus,  41- 
43- 

Descent,  Infinite,  86. 

Dickson,  81. 

Diophantine  equations,  84. 

Dirichlet,  81. 

Divisibility,  8. 

Divisors  of  a  numbers,  16,  17. 

Equations,  Diophantine,  84. 

Equation  xn+yn*=zn,  91-92. 

Eratosthenes,  n. 

Euclid,  Theorem  of,  13. 

Euclidian  algorithm,  18. 

Euler,  28,  48. 

Euler's  criterion,  59,  77. 

<£-function,  30. 
Exponent  of  an  integer,  61-63. 


Factorization  theorem,  14. 
Factors,  14,  16,  17,  18. 
Fermat,  28,  48,  86. 
Fermat's  general  theorem,  47,  63. 

last  theorem,  91. 

simple  theorem,  48,  55. 

theorem  extended,  52-54. 
Forms,  81-83. 
Fundamental  notions,  7. 

Galois  imaginaries,  80. 

Gauss,  37. 

Greatest  common  factor,  18-20,  21. 

Highest  power  of  p  in  n !,  24-28. 

Imaginaries  of  Galois,  80. 
Indicator,  30-36. 

of  any  integer,  32-34. 

of  a  prime  power,  30. 

of  a  product,  30-32'. 
Infinite  descent,  86. 

Mm),  53. 

Law  of  quadratic  reciprocity,  80. 
Least  common  multiple,  20-21. 
Legendre  symbol,  77. 

0O),  30- 

Prime  each  to  each,  9. 

Prime  numbers,  10,  12,  13,  28-29,  51) 

"76,  81,  82. 
Primitive  roots,  61-75. 
X-roots,  71-74. 
<]>-roots,  71. 
Pythagorean  triangles,  85-90. 

93 


94 


INDEX 


Quadratic  forms,  82. 
Quadratic  reciprocity,  80. 
Quadratic  residues,  57-60,  77-80. 

Relatively  prime,  10. 
Residue,  37,  58. 

Scales  of  notation,  22-24. 
Sieve  of  Eratosthenes,  10. 


Totient,  30. 

Triangles,  Numerical,  85. 

Unit,  8. 

Veblen,  81. 

Wilson's  theorem,  49-81. 

Young,  81. 


r?////5 


3     3S 


4^47 


7    3. 


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